Research Article On Hardy-Pachpatte-Copson s...
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Research Article On Hardy-Pachpatte-Copson’s Inequalities Chang-Jian Zhao 1 and Wing-Sum Cheung 2 1 Department of Mathematics, China Jiliang University, Hangzhou 310018, China 2 Department of Mathematics, e University of Hong Kong, Pokfulam Road, Hong Kong Correspondence should be addressed to Chang-Jian Zhao; [email protected] Received 6 December 2013; Accepted 19 January 2014; Published 18 March 2014 Academic Editors: B. Dragovich and D. Xu Copyright © 2014 C.-J. Zhao and W.-S. Cheung. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We establish new inequalities similar to Hardy-Pachpatte-Copson’s type inequalities. ese results in special cases yield some of the recent results. 1. Introduction e classical Hardy’s integral inequality is as follows. eorem A. If >1, () ≥ 0 for 0<<∞, and () = (1/) ∫ 0 (), then ∫ ∞ 0 () < ( −1 ) ∫ ∞ 0 () , (1) unless ≡0. e constant is the best possible. eorem A was first proved by Hardy [1], in an attempt to give a simple proof of Hilbert’s double series theorem (see [2]). One of the best known and interesting generalization of the inequality (1) given by Hardy [3] himself can be stated as follows. eorem B. If > 1, ̸ =1, () ≥ 0 for 0<<∞, and () is defined by () = ∫ 0 () , > 1; () = ∫ ∞ () , < 1, (2) then ∫ ∞ 0 − () < ( | − 1| ) ∫ ∞ 0 − () , (3) unless ≡0. e constant is the best possible. Inequalities (1) and (3) which later went by the name of Hardy’s inequalities led to a great many papers dealing with alternative proofs, various generalizations, and numer- ous variants and applications in analysis (see [4–15]). In particular, Pachpatte [4] established some generalizations of Hardy inequalities (1) and (3). Very recently, Leng and Feng [16] proved some new Hardy-type integral inequalities. In the present paper we establish new inequalities similar to Hardy’s integral inequalities (1) and (3). ese results provide some new estimates to these types of inequalities and in special cases yield some of the recent results. 2. Main Results Our main results are given in the following theorems. eorem 1. Let <<,<< ,>1,<1, and >0 be constants. Let (, ) be positive and locally absolutely continuous in (, ) × (, ). Let ℎ(, ) be a positive continuous function and let (, ) = ∫ ∫ ℎ(, ) , Hindawi Publishing Corporation e Scientific World Journal Volume 2014, Article ID 607347, 7 pages http://dx.doi.org/10.1155/2014/607347
Transcript of Research Article On Hardy-Pachpatte-Copson s...
Research ArticleOn Hardy-Pachpatte-Copsonrsquos Inequalities
Chang-Jian Zhao1 and Wing-Sum Cheung2
1 Department of Mathematics China Jiliang University Hangzhou 310018 China2Department of Mathematics The University of Hong Kong Pokfulam Road Hong Kong
Correspondence should be addressed to Chang-Jian Zhao chjzhaoaliyuncom
Received 6 December 2013 Accepted 19 January 2014 Published 18 March 2014
Academic Editors B Dragovich and D Xu
Copyright copy 2014 C-J Zhao and W-S CheungThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited
We establish new inequalities similar to Hardy-Pachpatte-Copsonrsquos type inequalities These results in special cases yield some ofthe recent results
1 Introduction
The classical Hardyrsquos integral inequality is as follows
Theorem A If 119901 gt 1 119891(119909) ge 0 for 0 lt 119909 lt infin and 119865(119909) =
(1119909) int119909
0
119891(119905)119889119905 then
int
infin
0
119865(119909)119901
119889119909 lt (119901
119901 minus 1)
119901
int
infin
0
119891(119909)119901
119889119909 (1)
unless 119891 equiv 0 The constant is the best possible
Theorem A was first proved by Hardy [1] in an attemptto give a simple proof of Hilbertrsquos double series theorem (see[2]) One of the best known and interesting generalization ofthe inequality (1) given by Hardy [3] himself can be stated asfollows
Theorem B If 119901 gt 1 119898 = 1 119891(119909) ge 0 for 0 lt 119909 lt infin and119865(119909) is defined by
119865 (119909) = int
119909
0
119891 (119905) 119889119905 119898 gt 1
119865 (119909) = int
infin
119909
119891 (119905) 119889119905 119898 lt 1
(2)
then
int
infin
0
119909minus119898
119865(119909)119901
119889119909 lt (119901
|119898 minus 1|)
119901
int
infin
0
119909119901minus119898
119891(119909)119901
119889119909 (3)
unless 119891 equiv 0 The constant is the best possible
Inequalities (1) and (3) which later went by the nameof Hardyrsquos inequalities led to a great many papers dealingwith alternative proofs various generalizations and numer-ous variants and applications in analysis (see [4ndash15]) Inparticular Pachpatte [4] established some generalizations ofHardy inequalities (1) and (3) Very recently Leng and Feng[16] proved some newHardy-type integral inequalities In thepresent paper we establish new inequalities similar to Hardyrsquosintegral inequalities (1) and (3) These results provide somenew estimates to these types of inequalities and in specialcases yield some of the recent results
2 Main Results
Our main results are given in the following theorems
Theorem 1 Let 119886 lt 119887 lt 119877 119888 lt 119889 lt 1198771015840
119901 gt 1 119902 lt 1and 120572 gt 0 be constants Let 119908(119909 119910) be positive and locallyabsolutely continuous in (119886 119887) times (119888 119889) Let ℎ(119909 119910) be a positivecontinuous function and let 119867(119909 119910) = int
119909
119886
int119910
119888
ℎ(119904 119905)119889119904 119889119905
Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 607347 7 pageshttpdxdoiorg1011552014607347
2 The Scientific World Journal
for (119909 119910) isin (119886 119887) times (119888 119889) Let 119891(119909 119910) be nonnegative andmeasurable on (119886 119887) times (119888 119889) If
119863(119909 119910)
= 1 minus1
1 minus 119902
119867 (119909 119910)
ℎ (119909 119910)
1
119908 (119909 119910)
120597119908 (119909 119910)
120597119909log(
119867(119877 1198771015840
)
119867 (119909 119910))
+119901
1 minus 119902
119867 (119909 119910)
ℎ (119909 119910)times
1
119903 (119909 119910)
120597119903 (119909 119910)
120597119909log(
119867(119877 1198771015840
)
119867 (119909 119910))
ge1
120572
(4)
for almost all (119909 119910) isin (119886 119887) times (119888 119889) and if 119865(119909 119910) is definedby
119865 (119909 119910) =1
119903 (119909 119910)int
119909
119886
int
119910
119888
119903 (119904 119905) ℎ (119904 119905) 119891 (119904 119905) 119889119904 119889119905 (5)
for (119909 119910) isin (119886 119887) times (119888 119889) then
int
119889
119888
int
119887
119886
119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times (log(119867(119877R1015840)119867 (119909 119910)
))
minus119902
119865(119909 119910)119901
119889119909 119889119910
le [120572(119901
1 minus 119902)]
119901
times int
119889
119888
int
119887
119886
[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
119901minus119902
119908 (119909 119910)119867(119909 119910)119901minus1
times ℎ(119909 119910)1minus119901
119866(119909 119910)119901]
]
119889119909119889119910
(6)
where
ℎ (119909 119910) = int
119910
119888
ℎ (119909 119905) 119889119905
119866 (119909 119910) =1
119903 (119909 119910)int
119910
119888
119903 (119909 119905) ℎ (119909 119905) 119891 (119909 119905) 119889119905
(7)
Remark 2 Let 119891(119909 119910) 119908(119909 119910) ℎ(119909 119910) and 119903(119909 119910) reduce to119891(119909) 119908(119909) ℎ(119909) and 119903(119909) respectively and with suitable
modifications in Theorem 1 (6) changes to the followingresult
int
119887
119886
119908 (119909)119867(119909)minus1
ℎ (119909) (log(119867 (119877)
119867 (119909)))
minus119902
119865(119909)119901
119889119909
le [120572(119901
1 minus 119902)]
119901
times int
119887
119886
[(log(119867 (119877)
119867 (119909)))
119901minus119902
times 119908 (119909)119867(119909)119901minus1
ℎ (119909) 119891(119909)119901
]119889119909
(8)
This is just a new inequality established by Pachpatte [4]Moreover we note that the inequality established in
Theorem 1 is the further generalizations of the inequalityestablished by Copson [17]
Taking for 119908(119909) = 119903(119909) = 1 119867(119877) = 119877 and 120572 = 1 in (8)(8) changes to the following result
int
119887
119886
119867(119909)minus1
ℎ (119909) (log( 119877
119867 (119909)))
minus119902
119865(119909)119901
119889119909
le (119901
1 minus 119902)
119901
times int
119887
119886
[(log( 119877
119867 (119909)))
119901minus119902
119867(119909)119901minus1
ℎ (119909) 119891(119909)119901
]119889119909
(9)
This is just a new inequality established by Love [7]Let ℎ(119909) = 1 119886 rarr 0 119887 rarr infin and log (119877(119909 minus 119886)) = 1
in (9) then (9) changes to the following result
int
infin
0
119909minus1
119865(119909)119901
119889119909 le (119901
1 minus 119902)
119901
int
infin
0
119909119901minus1
119891(119909)119901
119889119909 (10)
This result is obtained in (3) stated in the Introduction
Theorem 3 Let 119886 lt 119887 lt 119877 119888 lt 119889 lt 1198771015840
119901 gt 1 119902 gt 1 and 120573 gt
0 be constants Let 119908(119909 119910) be positive and locally absolutelycontinuous in (119886 119887)times(119888 119889) Let ℎ(119909 119910) be a positive continuousfunction and let 119867(119909 119910) = int
119909
119886
int119910
119888
ℎ(119904 119905)119889119904 119889119905 for (119909 119910) isin
(119886 119887) times (119888 119889) Let 119891(119909 119910) be nonnegative and measurable on(119886 119887) times (119888 119889) Let
119864 (119909 119910)
= 1 minus1
119902 minus 1
119867 (119909 119910)
ℎ (119909 119910)
1
119908 (119909 119910)
120597119908 (119909 119910)
120597119910log(
119867(119877 1198771015840
)
119867 (119909 119910))
+119901
119902 minus 1
119867 (119909 119910)
ℎ (119909 119910)times
1
119903 (119909 119910)
120597119903 (119909 119910)
120597119910log(
119867(119877 1198771015840
)
119867 (119909 119910))
ge1
120573
(11)
The Scientific World Journal 3
for almost all (119909 119910) isin (119886 119887) times (119888 119889) If 119865(119909 y) is defined by
119865 (119909 119910) =1
119903 (119909 119910)int
119887
119909
int
119889
119910
119903 (119904 119905) ℎ (119904 119905) 119891 (119904 119905) 119889119904 119889119905 (12)
for (119909 119910) isin (119886 119887) times (119888 119889) then
int
119889
119888
int
119887
119886
119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119865(119909 119910)119901
119889119909 119889119910
le [120573(119901
119902 minus 1)]
119901
times int
119889
119888
int
119887
119886
[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
119901minus119902
119908 (119909 119910)119867(119909 119910)119901minus1
times ℎ(119909 119910)1minus119901
119871(119909 119910)119901]
]
119889119909119889119910
(13)
where
ℎ (119909 119910) = int
119909
119886
ℎ (119904 119910) 119889119904
(14)
Remark 4 Let 119891(119909 119910) 119908(119909 119910) ℎ(119909 119910) and 119903(119909 119910) reduceto 119891(119909) 119908(119909) ℎ(119909) and 119903(119909) respectively and with suitablemodifications in Theorem 3 (13) changes to the followingresult
int
119887
119886
119908 (119909)119867(119909)minus1
ℎ (119909) (log(119867 (119877)
119867 (119909)))
minus119902
119865(119909)119901
119889119909
le [120573(119901
119902 minus 1)]
119901
int
119887
119886
[(log(119867(119877)
119867(119909)))
119901minus119902
times 119908 (119909)119867(119909)119901minus1
ℎ (119909) 119891(119909)119901
]119889119909
(15)
This is just a new inequality established by Pachpatte [4]On the other hand we note that the inequality established
in Theorem 3 is the further generalizations of the inequalityestablished by Copson [17]
Taking for119908(119909) = 119903(119909) = 1 119867(119877) = 119877 and 120573 = 1 in (15)(15) changes to the following result
int
119887
119886
119867(119909)minus1
ℎ (119909) (log( 119877
119867 (119909)))
minus119902
119865(119909)119901
119889119909
le (119901
119902 minus 1)
119901
int
119887
119886
[(log( 119877
119867 (119909)))
119901minus119902
times 119867(119909)119901minus1
ℎ (119909) 119891(119909)119901
]119889119909
(16)
This is just a new inequality established by Love [7]
3 Proof of Theorems
Proof of Theorem 1 If we let 119906(119909 119910) = 119908(119909 119910)119865(119909 119910)119901 and
in view of
119865 (119909 119910) =1
119903 (119909 119910)int
119909
119886
int
119910
119888
119903 (119904 119905) ℎ (119904 119905) 119891 (119904 119905) 119889119904 119889119905 (17)
for (119909 119910) isin (119886 119887) times (119888 119889) then
120597119906 (119909 119910)
120597119909
=120597119908 (119909 119910)
120597119909119865(119909 119910)
119901
+ 119908 (119909 119910) 119901119865(119909 119910)119901minus1
times (1
119903 (119909 119910)int
119910
119888
119903 (119909 119905) ℎ (119909 119905) 119891 (119909 119905) 119889119905
minus120597119903 (119909 119910) 120597119909
1199032 (119909 119910)
timesint
119909
119886
int
119910
119888
119903 (119904 119905) ℎ (119904 119905) 119891 (119904 119905) 119889119904 119889119905)
(18)
Let
120597V (119909 119910)120597119909
= 119867(119909 119910)minus1
ℎ (119909 119910) [log(119867(119877 119877
1015840
)
119867 (119909 119910))]
minus119902
(19)
where ℎ(119909 119910) = int119910
119888
ℎ(119909 119905)119889119905 and in view of 119867(119909 119910) =
int119909
119886
int119910
119888
ℎ(119904 119905)119889119904 119889119905 for (119909 119910) isin (119886 119887) times (119888 119889) then
V (119909 119910) = minus[log (119867 (119877 119877
1015840
) 119867 (119909 119910))]minus119902+1
(minus119902 + 1) (20)
4 The Scientific World Journal
From (18) (20) and integrating by parts for 119909 we have
int
119889
119888
int
119887
119886
119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119865(119909 119910)119901
119889119909 119889119910
= minusint
119889
119888
119908(119909 119910) 119865(119909 119910)119901
times[log (119867 (119877 119877
1015840
) 119867 (119909 119910))]minus119902+1
minus119902 + 1
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119909=119887
119909=119886
minus int
119887
119886
[log (119867 (119877 1198771015840
) 119867 (119909 119910))]minus119902+1
minus119902 + 1
times [120597119908 (119909 119910)
120597119909119865(119909 119910)
119901
+ 119908 (119909 119910) 119901119865(119909 119910)119901minus1
times (119866 (119909 119910) minus120597119903 (119909 119910) 120597119909
1199032 (119909 119910)
times int
119909
119886
int
119910
119888
119903 (119904 119905) ℎ (119904 119905)
times119891(119904119905)119889119904 119889119905)]119889119909
119889119910
(21)
where
119866 (119909 119910) =1
119903 (119909 119910)int
119910
119888
119903 (119909 119905) ℎ (119909 119905) 119891 (119909 119905) 119889119905 (22)
If 119902 lt 1 then we observe that
int
119889
119888
int
119887
119886
119863(119909 119910)119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119865(119909 119910)119901
119889119909 119889119910
le119901
1 minus 119902int
119889
119888
int
119887
119886
[
[
119908 (119909 119910)(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902+1
times 119866 (119909 119910) 119865(119909 119910)119901minus1]
]
119889119909119889119910
=119901
1 minus 119902
times int
119889
119888
int
119887
119886
[[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
1119901
times log(119867(119877 119877
1015840
)
119867 (119909 119910))119908(119909 119910)
1119901
times 119867(119909 119910)(119901minus1)119901
times ℎ(119909 119910)minus(119901minus1)119901
119866 (119909 119910)]]
]
times[[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
(119901minus1)119901
times 119908(119909 119910)(119901minus1)119901
times [119867(119909 119910)minus1
times ℎ (119909 119910)](119901minus1)119901
times 119865(119909 119910)119901minus1]
]
]
119889119909119889119910
(23)
By applying Holderrsquos inequality with indices 119901 119901(119901 minus 1) onthe right side of (23) we obtain
int
119889
119888
int
119887
119886
119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119865(119909 119910)119901
119889119909 119889119910
le 120572(119901
1 minus 119902)
times
int
119889
119888
int
119887
119886
[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
119901
times 119908 (119909 119910)119867(119909 119910)119901minus1
times ℎ(119909 119910)minus(119901minus1)
119866(119909 119910)119901]
]
119889119909119889119910
1119901
The Scientific World Journal 5
times
int
119889
119888
int
119887
119886
[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119908 (119909 119910)119867(119909 119910)minus1
times ℎ (119909 119910) 119865(119909 119910)119901]
]
119889119909119889119910
(119901minus1)119901
(24)
Dividing both sides of (24) by the second integral factor onthe right side of (24) and raising both sides to the 119901th powerwe obtain
int
119889
119888
int
119887
119886
119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119865(119909 119910)119901
119889119909 119889119910
le [120572(119901
1 minus 119902)]
119901
times int
119889
119888
int
119887
119886
[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
119901minus119902
times 119908 (119909 119910)119867(119909 119910)119901minus1
times ℎ(119909 119910)minus(119901minus1)
119866(119909 119910)119901]
]
119889119909119889119910
(25)
Proof of Theorem 3 If we let 119906(119909 119910) = 119908(119909 119910)119865(119909 119910)119901 and in
view of
119865 (119909 119910) =1
119903 (119909 119910)int
119887
119909
int
119889
119910
119903 (119904 119905) ℎ (119904 119905) 119891 (119904 119905) 119889119904 119889119905 (26)
for (119909 119910) isin (119886 119887) times (119888 119889) then
120597119906 (119909 119910)
120597119910
=120597119908 (119909 119910)
120597119910119865(119909 119910)
119901
+ 119908 (119909 119910) 119901119865(119909 119910)119901minus1
times (minus1
119903 (119909 119910)int
119889
119910
119903 (119909 119905) ℎ (119909 119905) 119891 (119909 119905) 119889119905
minus120597119903 (119909 119910) 120597119910
1199032 (119909 119910)int
119887
119909
int
119889
119910
119903 (119904 119905) ℎ (119904 119905) 119891 (119904 119905) 119889119904 119889119905)
(27)
Let
120597V (119909 119910)120597119910
= 119867(119909 119910)minus1
ℎ (119909 119910) [log(119867(119877 119877
1015840
)
119867 (119909 119910))]
minus119902
(28)
where ℎ(119909 119910) = int119909
119886
ℎ(119904 119910)119889119904 and in view of 119867(119909 119910) =
int119909
119886
int119910
119888
ℎ(119904 119905)119889119904 119889119905 for (119909 119910) isin (119886 119887) times (119888 119889) then
V (119909 119910) = minus[log (119867 (119877 119877
1015840
) 119867 (119909 119910))]minus119902+1
(minus119902 + 1) (29)
From (27) (29) and integrating by parts for 119910 we have
int
119889
119888
int
119887
119886
119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119865(119909 119910)119901
119889119909 119889119910
= minusint
119887
119886
119908(119909 119910) 119865(119909 119910)119901
times[log (119867 (119877 119877
1015840
) 119867 (119909 119910))]minus119902+1
minus119902 + 1
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119910=119889
119910=119888
minus int
119889
119888
[log (119867 (119877 1198771015840
) 119867 (119909 119910))]minus119902+1
minus119902 + 1
times [120597119908 (119909 119910)
120597119910119865(119909 119910)
119901
+ 119908 (119909 119910) 119901119865(119909 119910)119901minus1
times (119871 (119909 119910) minus120597119903 (119909 119910) 120597119910
1199032 (119909 119910)
times int
119887
119909
int
119889
119910
119903 (119904 119905) ℎ (119904 119905)
times119891(119904 119905)119889119904 119889119905)]119889119910
119889119909
(30)
where
119871 (119909 119910) = minus1
119903 (119909 119910)int
119887
119909
119903 (119904 119910) ℎ (119904 119910) 119891 (119904 119910) 119889119904 (31)
If 119902 gt 1 then we observe that
int
119889
119888
int
119887
119886
119864 (119909 119910)119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119865(119909 119910)119901
119889119909 119889119910
6 The Scientific World Journal
le119901
119902 minus 1int
119889
119888
int
119887
119886
[
[
119908 (119909 119910)(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902+1
times 119871 (119909 119910) 119865(119909 119910)119901minus1]
]
119889119909119889119910
=119901
119902 minus 1
times int
119889
119888
int
119887
119886
[[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
1119901
times log(119867(119877 119877
1015840
)
119867 (119909 119910))119908(119909 119910)
1119901
119867(119909 119910)(119901minus1)119901
times ℎ(119909 119910)minus(119901minus1)119901
119871 (119909 119910)]]
]
times[[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
(119901minus1)119901
times 119908(119909 119910)(119901minus1)119901
times [119867(119909 119910)minus1
times ℎ (119909 119910)](119901minus1)119901
times 119865(119909 119910)119901minus1]
]
]
119889119909119889119910
(32)
By applying Holderrsquos inequality with indices 119901 119901(119901 minus 1) onthe right side of (32) we obtain
int
119889
119888
int
119887
119886
119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119865(119909 119910)119901
119889119909 119889119910
le 120573(119901
119902 minus 1)
times
int
119889
119888
int
119887
119886
[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
119901
119908 (119909 119910)119867(119909 119910)119901minus1
times ℎ(119909 119910)minus(119901minus1)
119871(119909 119910)119901]
]
119889119909119889119910
1119901
times
int
119889
119888
int
119887
119886
[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
times 119908 (119909 119910)119867(119909 119910)minus1
times ℎ (119909 119910) 119865(119909 119910)119901]
]
119889119909119889119910
(119901minus1)119901
(33)
Dividing both sides of (33) by the second integral factor onthe right side of (33) and raising both sides to the 119901th powerwe obtain
int
119889
119888
int
119887
119886
119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
times 119865(119909 119910)119901
119889119909 119889119910
le [120573(119901
119902 minus 1)]
119901
times int
119889
119888
int
119887
119886
[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
119901minus119902
119908 (119909 119910)119867(119909 119910)119901minus1
times ℎ(119909 119910)minus(119901minus1)
119871(119909 119910)119901]
]
119889119909119889119910
(34)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Chang-Jian Zhaorsquos research is supported by National NaturalScience Foundation of China (11371334)Wing-sumCheungrsquosresearch is partially supported by HKU URG grant
References
[1] G H Hardy ldquoNote on a theorem of Hilbertrdquo MathematischeZeitschrift vol 6 no 3-4 pp 314ndash317 1920
[2] G H Hardy J E Littlewood and G Polya InequalitiesCambridge University Press Cambridge UK 1934
[3] G H Hardy ldquoNotes on some points in the integral calculusrdquoMessenger of Mathematics vol 57 pp 12ndash16 1928
The Scientific World Journal 7
[4] B G Pachpatte ldquoOn some generalizations of Hardyrsquos integralinequalityrdquo Journal of Mathematical Analysis and Applicationsvol 234 no 1 pp 15ndash30 1999
[5] B G Pachpatte ldquoOn Hardy type integral inequalities forfunctions of two variablesrdquo Demonstratio Mathematica vol 28no 2 pp 239ndash244 1995
[6] J E Pecaric andE R Love ldquoStillmore generalizations ofHardyrsquosinequalityrdquo Journal of the Australian Mathematical Society vol58 pp 1ndash11 1995
[7] E R Love ldquoGeneralizations of Hardys inequalityrdquo Proceedingsof the Royal Society of Edinburgh vol 100 pp 237ndash262 1985
[8] B C Yang I Brnetic M Krnic and J Pecaric ldquoGeneralizationof Hilbert and Hardy-Hilbert integral inequalitiesrdquo Mathemat-ical Inequalities amp Applications vol 8 pp 259ndash272 2005
[9] B Yang and L Debnath ldquoOn the extended Hardy-Hilbertrsquosinequalityrdquo Journal of Mathematical Analysis and Applicationsvol 272 no 1 pp 187ndash199 2002
[10] K Jichang and L Debnath ldquoOn new generalizations of Hilbertrsquosinequality and their applicationsrdquo Journal of MathematicalAnalysis and Applications vol 245 no 1 pp 248ndash265 2000
[11] M Z Sarikaya and H Yildirim ldquoSome Hardy type integralinequalitiesrdquo Journal of Inequalities in Pure and Applied Mathe-matics vol 7 no 5 article 178 2006
[12] C J Zhao and L Debnath ldquoSome new inverse type Hilbertintegral inequalitiesrdquo Journal of Mathematical Analysis andApplications vol 262 pp 411ndash418 2001
[13] V M Miklyukov and M K Vuorinen ldquoHardyrsquos inequality forw11199010-functions on riemannian manifoldsrdquo Proceedings of the
American Mathematical Society vol 127 no 9 pp 2745ndash27541999
[14] B Opic and A Kufner Hardy-Type Inequalities LongmanEssex UK 1990
[15] A Kufner and L E Persson Weighted Inequalities of HardyType World Scientific 2003
[16] T Leng and Y Feng ldquoOn Hardy-type integral inequalitiesrdquoApplied Mathematics and Mechanics vol 34 no 10 pp 1297ndash1304 2013
[17] E T Copson ldquoSome integral inequalitiesrdquo Proceedings of theRoyal Society of Edinburgh vol 75 pp 157ndash164 1976
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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OptimizationJournal of
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International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 The Scientific World Journal
for (119909 119910) isin (119886 119887) times (119888 119889) Let 119891(119909 119910) be nonnegative andmeasurable on (119886 119887) times (119888 119889) If
119863(119909 119910)
= 1 minus1
1 minus 119902
119867 (119909 119910)
ℎ (119909 119910)
1
119908 (119909 119910)
120597119908 (119909 119910)
120597119909log(
119867(119877 1198771015840
)
119867 (119909 119910))
+119901
1 minus 119902
119867 (119909 119910)
ℎ (119909 119910)times
1
119903 (119909 119910)
120597119903 (119909 119910)
120597119909log(
119867(119877 1198771015840
)
119867 (119909 119910))
ge1
120572
(4)
for almost all (119909 119910) isin (119886 119887) times (119888 119889) and if 119865(119909 119910) is definedby
119865 (119909 119910) =1
119903 (119909 119910)int
119909
119886
int
119910
119888
119903 (119904 119905) ℎ (119904 119905) 119891 (119904 119905) 119889119904 119889119905 (5)
for (119909 119910) isin (119886 119887) times (119888 119889) then
int
119889
119888
int
119887
119886
119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times (log(119867(119877R1015840)119867 (119909 119910)
))
minus119902
119865(119909 119910)119901
119889119909 119889119910
le [120572(119901
1 minus 119902)]
119901
times int
119889
119888
int
119887
119886
[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
119901minus119902
119908 (119909 119910)119867(119909 119910)119901minus1
times ℎ(119909 119910)1minus119901
119866(119909 119910)119901]
]
119889119909119889119910
(6)
where
ℎ (119909 119910) = int
119910
119888
ℎ (119909 119905) 119889119905
119866 (119909 119910) =1
119903 (119909 119910)int
119910
119888
119903 (119909 119905) ℎ (119909 119905) 119891 (119909 119905) 119889119905
(7)
Remark 2 Let 119891(119909 119910) 119908(119909 119910) ℎ(119909 119910) and 119903(119909 119910) reduce to119891(119909) 119908(119909) ℎ(119909) and 119903(119909) respectively and with suitable
modifications in Theorem 1 (6) changes to the followingresult
int
119887
119886
119908 (119909)119867(119909)minus1
ℎ (119909) (log(119867 (119877)
119867 (119909)))
minus119902
119865(119909)119901
119889119909
le [120572(119901
1 minus 119902)]
119901
times int
119887
119886
[(log(119867 (119877)
119867 (119909)))
119901minus119902
times 119908 (119909)119867(119909)119901minus1
ℎ (119909) 119891(119909)119901
]119889119909
(8)
This is just a new inequality established by Pachpatte [4]Moreover we note that the inequality established in
Theorem 1 is the further generalizations of the inequalityestablished by Copson [17]
Taking for 119908(119909) = 119903(119909) = 1 119867(119877) = 119877 and 120572 = 1 in (8)(8) changes to the following result
int
119887
119886
119867(119909)minus1
ℎ (119909) (log( 119877
119867 (119909)))
minus119902
119865(119909)119901
119889119909
le (119901
1 minus 119902)
119901
times int
119887
119886
[(log( 119877
119867 (119909)))
119901minus119902
119867(119909)119901minus1
ℎ (119909) 119891(119909)119901
]119889119909
(9)
This is just a new inequality established by Love [7]Let ℎ(119909) = 1 119886 rarr 0 119887 rarr infin and log (119877(119909 minus 119886)) = 1
in (9) then (9) changes to the following result
int
infin
0
119909minus1
119865(119909)119901
119889119909 le (119901
1 minus 119902)
119901
int
infin
0
119909119901minus1
119891(119909)119901
119889119909 (10)
This result is obtained in (3) stated in the Introduction
Theorem 3 Let 119886 lt 119887 lt 119877 119888 lt 119889 lt 1198771015840
119901 gt 1 119902 gt 1 and 120573 gt
0 be constants Let 119908(119909 119910) be positive and locally absolutelycontinuous in (119886 119887)times(119888 119889) Let ℎ(119909 119910) be a positive continuousfunction and let 119867(119909 119910) = int
119909
119886
int119910
119888
ℎ(119904 119905)119889119904 119889119905 for (119909 119910) isin
(119886 119887) times (119888 119889) Let 119891(119909 119910) be nonnegative and measurable on(119886 119887) times (119888 119889) Let
119864 (119909 119910)
= 1 minus1
119902 minus 1
119867 (119909 119910)
ℎ (119909 119910)
1
119908 (119909 119910)
120597119908 (119909 119910)
120597119910log(
119867(119877 1198771015840
)
119867 (119909 119910))
+119901
119902 minus 1
119867 (119909 119910)
ℎ (119909 119910)times
1
119903 (119909 119910)
120597119903 (119909 119910)
120597119910log(
119867(119877 1198771015840
)
119867 (119909 119910))
ge1
120573
(11)
The Scientific World Journal 3
for almost all (119909 119910) isin (119886 119887) times (119888 119889) If 119865(119909 y) is defined by
119865 (119909 119910) =1
119903 (119909 119910)int
119887
119909
int
119889
119910
119903 (119904 119905) ℎ (119904 119905) 119891 (119904 119905) 119889119904 119889119905 (12)
for (119909 119910) isin (119886 119887) times (119888 119889) then
int
119889
119888
int
119887
119886
119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119865(119909 119910)119901
119889119909 119889119910
le [120573(119901
119902 minus 1)]
119901
times int
119889
119888
int
119887
119886
[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
119901minus119902
119908 (119909 119910)119867(119909 119910)119901minus1
times ℎ(119909 119910)1minus119901
119871(119909 119910)119901]
]
119889119909119889119910
(13)
where
ℎ (119909 119910) = int
119909
119886
ℎ (119904 119910) 119889119904
(14)
Remark 4 Let 119891(119909 119910) 119908(119909 119910) ℎ(119909 119910) and 119903(119909 119910) reduceto 119891(119909) 119908(119909) ℎ(119909) and 119903(119909) respectively and with suitablemodifications in Theorem 3 (13) changes to the followingresult
int
119887
119886
119908 (119909)119867(119909)minus1
ℎ (119909) (log(119867 (119877)
119867 (119909)))
minus119902
119865(119909)119901
119889119909
le [120573(119901
119902 minus 1)]
119901
int
119887
119886
[(log(119867(119877)
119867(119909)))
119901minus119902
times 119908 (119909)119867(119909)119901minus1
ℎ (119909) 119891(119909)119901
]119889119909
(15)
This is just a new inequality established by Pachpatte [4]On the other hand we note that the inequality established
in Theorem 3 is the further generalizations of the inequalityestablished by Copson [17]
Taking for119908(119909) = 119903(119909) = 1 119867(119877) = 119877 and 120573 = 1 in (15)(15) changes to the following result
int
119887
119886
119867(119909)minus1
ℎ (119909) (log( 119877
119867 (119909)))
minus119902
119865(119909)119901
119889119909
le (119901
119902 minus 1)
119901
int
119887
119886
[(log( 119877
119867 (119909)))
119901minus119902
times 119867(119909)119901minus1
ℎ (119909) 119891(119909)119901
]119889119909
(16)
This is just a new inequality established by Love [7]
3 Proof of Theorems
Proof of Theorem 1 If we let 119906(119909 119910) = 119908(119909 119910)119865(119909 119910)119901 and
in view of
119865 (119909 119910) =1
119903 (119909 119910)int
119909
119886
int
119910
119888
119903 (119904 119905) ℎ (119904 119905) 119891 (119904 119905) 119889119904 119889119905 (17)
for (119909 119910) isin (119886 119887) times (119888 119889) then
120597119906 (119909 119910)
120597119909
=120597119908 (119909 119910)
120597119909119865(119909 119910)
119901
+ 119908 (119909 119910) 119901119865(119909 119910)119901minus1
times (1
119903 (119909 119910)int
119910
119888
119903 (119909 119905) ℎ (119909 119905) 119891 (119909 119905) 119889119905
minus120597119903 (119909 119910) 120597119909
1199032 (119909 119910)
timesint
119909
119886
int
119910
119888
119903 (119904 119905) ℎ (119904 119905) 119891 (119904 119905) 119889119904 119889119905)
(18)
Let
120597V (119909 119910)120597119909
= 119867(119909 119910)minus1
ℎ (119909 119910) [log(119867(119877 119877
1015840
)
119867 (119909 119910))]
minus119902
(19)
where ℎ(119909 119910) = int119910
119888
ℎ(119909 119905)119889119905 and in view of 119867(119909 119910) =
int119909
119886
int119910
119888
ℎ(119904 119905)119889119904 119889119905 for (119909 119910) isin (119886 119887) times (119888 119889) then
V (119909 119910) = minus[log (119867 (119877 119877
1015840
) 119867 (119909 119910))]minus119902+1
(minus119902 + 1) (20)
4 The Scientific World Journal
From (18) (20) and integrating by parts for 119909 we have
int
119889
119888
int
119887
119886
119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119865(119909 119910)119901
119889119909 119889119910
= minusint
119889
119888
119908(119909 119910) 119865(119909 119910)119901
times[log (119867 (119877 119877
1015840
) 119867 (119909 119910))]minus119902+1
minus119902 + 1
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119909=119887
119909=119886
minus int
119887
119886
[log (119867 (119877 1198771015840
) 119867 (119909 119910))]minus119902+1
minus119902 + 1
times [120597119908 (119909 119910)
120597119909119865(119909 119910)
119901
+ 119908 (119909 119910) 119901119865(119909 119910)119901minus1
times (119866 (119909 119910) minus120597119903 (119909 119910) 120597119909
1199032 (119909 119910)
times int
119909
119886
int
119910
119888
119903 (119904 119905) ℎ (119904 119905)
times119891(119904119905)119889119904 119889119905)]119889119909
119889119910
(21)
where
119866 (119909 119910) =1
119903 (119909 119910)int
119910
119888
119903 (119909 119905) ℎ (119909 119905) 119891 (119909 119905) 119889119905 (22)
If 119902 lt 1 then we observe that
int
119889
119888
int
119887
119886
119863(119909 119910)119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119865(119909 119910)119901
119889119909 119889119910
le119901
1 minus 119902int
119889
119888
int
119887
119886
[
[
119908 (119909 119910)(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902+1
times 119866 (119909 119910) 119865(119909 119910)119901minus1]
]
119889119909119889119910
=119901
1 minus 119902
times int
119889
119888
int
119887
119886
[[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
1119901
times log(119867(119877 119877
1015840
)
119867 (119909 119910))119908(119909 119910)
1119901
times 119867(119909 119910)(119901minus1)119901
times ℎ(119909 119910)minus(119901minus1)119901
119866 (119909 119910)]]
]
times[[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
(119901minus1)119901
times 119908(119909 119910)(119901minus1)119901
times [119867(119909 119910)minus1
times ℎ (119909 119910)](119901minus1)119901
times 119865(119909 119910)119901minus1]
]
]
119889119909119889119910
(23)
By applying Holderrsquos inequality with indices 119901 119901(119901 minus 1) onthe right side of (23) we obtain
int
119889
119888
int
119887
119886
119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119865(119909 119910)119901
119889119909 119889119910
le 120572(119901
1 minus 119902)
times
int
119889
119888
int
119887
119886
[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
119901
times 119908 (119909 119910)119867(119909 119910)119901minus1
times ℎ(119909 119910)minus(119901minus1)
119866(119909 119910)119901]
]
119889119909119889119910
1119901
The Scientific World Journal 5
times
int
119889
119888
int
119887
119886
[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119908 (119909 119910)119867(119909 119910)minus1
times ℎ (119909 119910) 119865(119909 119910)119901]
]
119889119909119889119910
(119901minus1)119901
(24)
Dividing both sides of (24) by the second integral factor onthe right side of (24) and raising both sides to the 119901th powerwe obtain
int
119889
119888
int
119887
119886
119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119865(119909 119910)119901
119889119909 119889119910
le [120572(119901
1 minus 119902)]
119901
times int
119889
119888
int
119887
119886
[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
119901minus119902
times 119908 (119909 119910)119867(119909 119910)119901minus1
times ℎ(119909 119910)minus(119901minus1)
119866(119909 119910)119901]
]
119889119909119889119910
(25)
Proof of Theorem 3 If we let 119906(119909 119910) = 119908(119909 119910)119865(119909 119910)119901 and in
view of
119865 (119909 119910) =1
119903 (119909 119910)int
119887
119909
int
119889
119910
119903 (119904 119905) ℎ (119904 119905) 119891 (119904 119905) 119889119904 119889119905 (26)
for (119909 119910) isin (119886 119887) times (119888 119889) then
120597119906 (119909 119910)
120597119910
=120597119908 (119909 119910)
120597119910119865(119909 119910)
119901
+ 119908 (119909 119910) 119901119865(119909 119910)119901minus1
times (minus1
119903 (119909 119910)int
119889
119910
119903 (119909 119905) ℎ (119909 119905) 119891 (119909 119905) 119889119905
minus120597119903 (119909 119910) 120597119910
1199032 (119909 119910)int
119887
119909
int
119889
119910
119903 (119904 119905) ℎ (119904 119905) 119891 (119904 119905) 119889119904 119889119905)
(27)
Let
120597V (119909 119910)120597119910
= 119867(119909 119910)minus1
ℎ (119909 119910) [log(119867(119877 119877
1015840
)
119867 (119909 119910))]
minus119902
(28)
where ℎ(119909 119910) = int119909
119886
ℎ(119904 119910)119889119904 and in view of 119867(119909 119910) =
int119909
119886
int119910
119888
ℎ(119904 119905)119889119904 119889119905 for (119909 119910) isin (119886 119887) times (119888 119889) then
V (119909 119910) = minus[log (119867 (119877 119877
1015840
) 119867 (119909 119910))]minus119902+1
(minus119902 + 1) (29)
From (27) (29) and integrating by parts for 119910 we have
int
119889
119888
int
119887
119886
119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119865(119909 119910)119901
119889119909 119889119910
= minusint
119887
119886
119908(119909 119910) 119865(119909 119910)119901
times[log (119867 (119877 119877
1015840
) 119867 (119909 119910))]minus119902+1
minus119902 + 1
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119910=119889
119910=119888
minus int
119889
119888
[log (119867 (119877 1198771015840
) 119867 (119909 119910))]minus119902+1
minus119902 + 1
times [120597119908 (119909 119910)
120597119910119865(119909 119910)
119901
+ 119908 (119909 119910) 119901119865(119909 119910)119901minus1
times (119871 (119909 119910) minus120597119903 (119909 119910) 120597119910
1199032 (119909 119910)
times int
119887
119909
int
119889
119910
119903 (119904 119905) ℎ (119904 119905)
times119891(119904 119905)119889119904 119889119905)]119889119910
119889119909
(30)
where
119871 (119909 119910) = minus1
119903 (119909 119910)int
119887
119909
119903 (119904 119910) ℎ (119904 119910) 119891 (119904 119910) 119889119904 (31)
If 119902 gt 1 then we observe that
int
119889
119888
int
119887
119886
119864 (119909 119910)119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119865(119909 119910)119901
119889119909 119889119910
6 The Scientific World Journal
le119901
119902 minus 1int
119889
119888
int
119887
119886
[
[
119908 (119909 119910)(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902+1
times 119871 (119909 119910) 119865(119909 119910)119901minus1]
]
119889119909119889119910
=119901
119902 minus 1
times int
119889
119888
int
119887
119886
[[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
1119901
times log(119867(119877 119877
1015840
)
119867 (119909 119910))119908(119909 119910)
1119901
119867(119909 119910)(119901minus1)119901
times ℎ(119909 119910)minus(119901minus1)119901
119871 (119909 119910)]]
]
times[[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
(119901minus1)119901
times 119908(119909 119910)(119901minus1)119901
times [119867(119909 119910)minus1
times ℎ (119909 119910)](119901minus1)119901
times 119865(119909 119910)119901minus1]
]
]
119889119909119889119910
(32)
By applying Holderrsquos inequality with indices 119901 119901(119901 minus 1) onthe right side of (32) we obtain
int
119889
119888
int
119887
119886
119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119865(119909 119910)119901
119889119909 119889119910
le 120573(119901
119902 minus 1)
times
int
119889
119888
int
119887
119886
[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
119901
119908 (119909 119910)119867(119909 119910)119901minus1
times ℎ(119909 119910)minus(119901minus1)
119871(119909 119910)119901]
]
119889119909119889119910
1119901
times
int
119889
119888
int
119887
119886
[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
times 119908 (119909 119910)119867(119909 119910)minus1
times ℎ (119909 119910) 119865(119909 119910)119901]
]
119889119909119889119910
(119901minus1)119901
(33)
Dividing both sides of (33) by the second integral factor onthe right side of (33) and raising both sides to the 119901th powerwe obtain
int
119889
119888
int
119887
119886
119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
times 119865(119909 119910)119901
119889119909 119889119910
le [120573(119901
119902 minus 1)]
119901
times int
119889
119888
int
119887
119886
[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
119901minus119902
119908 (119909 119910)119867(119909 119910)119901minus1
times ℎ(119909 119910)minus(119901minus1)
119871(119909 119910)119901]
]
119889119909119889119910
(34)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Chang-Jian Zhaorsquos research is supported by National NaturalScience Foundation of China (11371334)Wing-sumCheungrsquosresearch is partially supported by HKU URG grant
References
[1] G H Hardy ldquoNote on a theorem of Hilbertrdquo MathematischeZeitschrift vol 6 no 3-4 pp 314ndash317 1920
[2] G H Hardy J E Littlewood and G Polya InequalitiesCambridge University Press Cambridge UK 1934
[3] G H Hardy ldquoNotes on some points in the integral calculusrdquoMessenger of Mathematics vol 57 pp 12ndash16 1928
The Scientific World Journal 7
[4] B G Pachpatte ldquoOn some generalizations of Hardyrsquos integralinequalityrdquo Journal of Mathematical Analysis and Applicationsvol 234 no 1 pp 15ndash30 1999
[5] B G Pachpatte ldquoOn Hardy type integral inequalities forfunctions of two variablesrdquo Demonstratio Mathematica vol 28no 2 pp 239ndash244 1995
[6] J E Pecaric andE R Love ldquoStillmore generalizations ofHardyrsquosinequalityrdquo Journal of the Australian Mathematical Society vol58 pp 1ndash11 1995
[7] E R Love ldquoGeneralizations of Hardys inequalityrdquo Proceedingsof the Royal Society of Edinburgh vol 100 pp 237ndash262 1985
[8] B C Yang I Brnetic M Krnic and J Pecaric ldquoGeneralizationof Hilbert and Hardy-Hilbert integral inequalitiesrdquo Mathemat-ical Inequalities amp Applications vol 8 pp 259ndash272 2005
[9] B Yang and L Debnath ldquoOn the extended Hardy-Hilbertrsquosinequalityrdquo Journal of Mathematical Analysis and Applicationsvol 272 no 1 pp 187ndash199 2002
[10] K Jichang and L Debnath ldquoOn new generalizations of Hilbertrsquosinequality and their applicationsrdquo Journal of MathematicalAnalysis and Applications vol 245 no 1 pp 248ndash265 2000
[11] M Z Sarikaya and H Yildirim ldquoSome Hardy type integralinequalitiesrdquo Journal of Inequalities in Pure and Applied Mathe-matics vol 7 no 5 article 178 2006
[12] C J Zhao and L Debnath ldquoSome new inverse type Hilbertintegral inequalitiesrdquo Journal of Mathematical Analysis andApplications vol 262 pp 411ndash418 2001
[13] V M Miklyukov and M K Vuorinen ldquoHardyrsquos inequality forw11199010-functions on riemannian manifoldsrdquo Proceedings of the
American Mathematical Society vol 127 no 9 pp 2745ndash27541999
[14] B Opic and A Kufner Hardy-Type Inequalities LongmanEssex UK 1990
[15] A Kufner and L E Persson Weighted Inequalities of HardyType World Scientific 2003
[16] T Leng and Y Feng ldquoOn Hardy-type integral inequalitiesrdquoApplied Mathematics and Mechanics vol 34 no 10 pp 1297ndash1304 2013
[17] E T Copson ldquoSome integral inequalitiesrdquo Proceedings of theRoyal Society of Edinburgh vol 75 pp 157ndash164 1976
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
The Scientific World Journal 3
for almost all (119909 119910) isin (119886 119887) times (119888 119889) If 119865(119909 y) is defined by
119865 (119909 119910) =1
119903 (119909 119910)int
119887
119909
int
119889
119910
119903 (119904 119905) ℎ (119904 119905) 119891 (119904 119905) 119889119904 119889119905 (12)
for (119909 119910) isin (119886 119887) times (119888 119889) then
int
119889
119888
int
119887
119886
119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119865(119909 119910)119901
119889119909 119889119910
le [120573(119901
119902 minus 1)]
119901
times int
119889
119888
int
119887
119886
[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
119901minus119902
119908 (119909 119910)119867(119909 119910)119901minus1
times ℎ(119909 119910)1minus119901
119871(119909 119910)119901]
]
119889119909119889119910
(13)
where
ℎ (119909 119910) = int
119909
119886
ℎ (119904 119910) 119889119904
(14)
Remark 4 Let 119891(119909 119910) 119908(119909 119910) ℎ(119909 119910) and 119903(119909 119910) reduceto 119891(119909) 119908(119909) ℎ(119909) and 119903(119909) respectively and with suitablemodifications in Theorem 3 (13) changes to the followingresult
int
119887
119886
119908 (119909)119867(119909)minus1
ℎ (119909) (log(119867 (119877)
119867 (119909)))
minus119902
119865(119909)119901
119889119909
le [120573(119901
119902 minus 1)]
119901
int
119887
119886
[(log(119867(119877)
119867(119909)))
119901minus119902
times 119908 (119909)119867(119909)119901minus1
ℎ (119909) 119891(119909)119901
]119889119909
(15)
This is just a new inequality established by Pachpatte [4]On the other hand we note that the inequality established
in Theorem 3 is the further generalizations of the inequalityestablished by Copson [17]
Taking for119908(119909) = 119903(119909) = 1 119867(119877) = 119877 and 120573 = 1 in (15)(15) changes to the following result
int
119887
119886
119867(119909)minus1
ℎ (119909) (log( 119877
119867 (119909)))
minus119902
119865(119909)119901
119889119909
le (119901
119902 minus 1)
119901
int
119887
119886
[(log( 119877
119867 (119909)))
119901minus119902
times 119867(119909)119901minus1
ℎ (119909) 119891(119909)119901
]119889119909
(16)
This is just a new inequality established by Love [7]
3 Proof of Theorems
Proof of Theorem 1 If we let 119906(119909 119910) = 119908(119909 119910)119865(119909 119910)119901 and
in view of
119865 (119909 119910) =1
119903 (119909 119910)int
119909
119886
int
119910
119888
119903 (119904 119905) ℎ (119904 119905) 119891 (119904 119905) 119889119904 119889119905 (17)
for (119909 119910) isin (119886 119887) times (119888 119889) then
120597119906 (119909 119910)
120597119909
=120597119908 (119909 119910)
120597119909119865(119909 119910)
119901
+ 119908 (119909 119910) 119901119865(119909 119910)119901minus1
times (1
119903 (119909 119910)int
119910
119888
119903 (119909 119905) ℎ (119909 119905) 119891 (119909 119905) 119889119905
minus120597119903 (119909 119910) 120597119909
1199032 (119909 119910)
timesint
119909
119886
int
119910
119888
119903 (119904 119905) ℎ (119904 119905) 119891 (119904 119905) 119889119904 119889119905)
(18)
Let
120597V (119909 119910)120597119909
= 119867(119909 119910)minus1
ℎ (119909 119910) [log(119867(119877 119877
1015840
)
119867 (119909 119910))]
minus119902
(19)
where ℎ(119909 119910) = int119910
119888
ℎ(119909 119905)119889119905 and in view of 119867(119909 119910) =
int119909
119886
int119910
119888
ℎ(119904 119905)119889119904 119889119905 for (119909 119910) isin (119886 119887) times (119888 119889) then
V (119909 119910) = minus[log (119867 (119877 119877
1015840
) 119867 (119909 119910))]minus119902+1
(minus119902 + 1) (20)
4 The Scientific World Journal
From (18) (20) and integrating by parts for 119909 we have
int
119889
119888
int
119887
119886
119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119865(119909 119910)119901
119889119909 119889119910
= minusint
119889
119888
119908(119909 119910) 119865(119909 119910)119901
times[log (119867 (119877 119877
1015840
) 119867 (119909 119910))]minus119902+1
minus119902 + 1
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119909=119887
119909=119886
minus int
119887
119886
[log (119867 (119877 1198771015840
) 119867 (119909 119910))]minus119902+1
minus119902 + 1
times [120597119908 (119909 119910)
120597119909119865(119909 119910)
119901
+ 119908 (119909 119910) 119901119865(119909 119910)119901minus1
times (119866 (119909 119910) minus120597119903 (119909 119910) 120597119909
1199032 (119909 119910)
times int
119909
119886
int
119910
119888
119903 (119904 119905) ℎ (119904 119905)
times119891(119904119905)119889119904 119889119905)]119889119909
119889119910
(21)
where
119866 (119909 119910) =1
119903 (119909 119910)int
119910
119888
119903 (119909 119905) ℎ (119909 119905) 119891 (119909 119905) 119889119905 (22)
If 119902 lt 1 then we observe that
int
119889
119888
int
119887
119886
119863(119909 119910)119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119865(119909 119910)119901
119889119909 119889119910
le119901
1 minus 119902int
119889
119888
int
119887
119886
[
[
119908 (119909 119910)(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902+1
times 119866 (119909 119910) 119865(119909 119910)119901minus1]
]
119889119909119889119910
=119901
1 minus 119902
times int
119889
119888
int
119887
119886
[[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
1119901
times log(119867(119877 119877
1015840
)
119867 (119909 119910))119908(119909 119910)
1119901
times 119867(119909 119910)(119901minus1)119901
times ℎ(119909 119910)minus(119901minus1)119901
119866 (119909 119910)]]
]
times[[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
(119901minus1)119901
times 119908(119909 119910)(119901minus1)119901
times [119867(119909 119910)minus1
times ℎ (119909 119910)](119901minus1)119901
times 119865(119909 119910)119901minus1]
]
]
119889119909119889119910
(23)
By applying Holderrsquos inequality with indices 119901 119901(119901 minus 1) onthe right side of (23) we obtain
int
119889
119888
int
119887
119886
119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119865(119909 119910)119901
119889119909 119889119910
le 120572(119901
1 minus 119902)
times
int
119889
119888
int
119887
119886
[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
119901
times 119908 (119909 119910)119867(119909 119910)119901minus1
times ℎ(119909 119910)minus(119901minus1)
119866(119909 119910)119901]
]
119889119909119889119910
1119901
The Scientific World Journal 5
times
int
119889
119888
int
119887
119886
[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119908 (119909 119910)119867(119909 119910)minus1
times ℎ (119909 119910) 119865(119909 119910)119901]
]
119889119909119889119910
(119901minus1)119901
(24)
Dividing both sides of (24) by the second integral factor onthe right side of (24) and raising both sides to the 119901th powerwe obtain
int
119889
119888
int
119887
119886
119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119865(119909 119910)119901
119889119909 119889119910
le [120572(119901
1 minus 119902)]
119901
times int
119889
119888
int
119887
119886
[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
119901minus119902
times 119908 (119909 119910)119867(119909 119910)119901minus1
times ℎ(119909 119910)minus(119901minus1)
119866(119909 119910)119901]
]
119889119909119889119910
(25)
Proof of Theorem 3 If we let 119906(119909 119910) = 119908(119909 119910)119865(119909 119910)119901 and in
view of
119865 (119909 119910) =1
119903 (119909 119910)int
119887
119909
int
119889
119910
119903 (119904 119905) ℎ (119904 119905) 119891 (119904 119905) 119889119904 119889119905 (26)
for (119909 119910) isin (119886 119887) times (119888 119889) then
120597119906 (119909 119910)
120597119910
=120597119908 (119909 119910)
120597119910119865(119909 119910)
119901
+ 119908 (119909 119910) 119901119865(119909 119910)119901minus1
times (minus1
119903 (119909 119910)int
119889
119910
119903 (119909 119905) ℎ (119909 119905) 119891 (119909 119905) 119889119905
minus120597119903 (119909 119910) 120597119910
1199032 (119909 119910)int
119887
119909
int
119889
119910
119903 (119904 119905) ℎ (119904 119905) 119891 (119904 119905) 119889119904 119889119905)
(27)
Let
120597V (119909 119910)120597119910
= 119867(119909 119910)minus1
ℎ (119909 119910) [log(119867(119877 119877
1015840
)
119867 (119909 119910))]
minus119902
(28)
where ℎ(119909 119910) = int119909
119886
ℎ(119904 119910)119889119904 and in view of 119867(119909 119910) =
int119909
119886
int119910
119888
ℎ(119904 119905)119889119904 119889119905 for (119909 119910) isin (119886 119887) times (119888 119889) then
V (119909 119910) = minus[log (119867 (119877 119877
1015840
) 119867 (119909 119910))]minus119902+1
(minus119902 + 1) (29)
From (27) (29) and integrating by parts for 119910 we have
int
119889
119888
int
119887
119886
119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119865(119909 119910)119901
119889119909 119889119910
= minusint
119887
119886
119908(119909 119910) 119865(119909 119910)119901
times[log (119867 (119877 119877
1015840
) 119867 (119909 119910))]minus119902+1
minus119902 + 1
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119910=119889
119910=119888
minus int
119889
119888
[log (119867 (119877 1198771015840
) 119867 (119909 119910))]minus119902+1
minus119902 + 1
times [120597119908 (119909 119910)
120597119910119865(119909 119910)
119901
+ 119908 (119909 119910) 119901119865(119909 119910)119901minus1
times (119871 (119909 119910) minus120597119903 (119909 119910) 120597119910
1199032 (119909 119910)
times int
119887
119909
int
119889
119910
119903 (119904 119905) ℎ (119904 119905)
times119891(119904 119905)119889119904 119889119905)]119889119910
119889119909
(30)
where
119871 (119909 119910) = minus1
119903 (119909 119910)int
119887
119909
119903 (119904 119910) ℎ (119904 119910) 119891 (119904 119910) 119889119904 (31)
If 119902 gt 1 then we observe that
int
119889
119888
int
119887
119886
119864 (119909 119910)119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119865(119909 119910)119901
119889119909 119889119910
6 The Scientific World Journal
le119901
119902 minus 1int
119889
119888
int
119887
119886
[
[
119908 (119909 119910)(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902+1
times 119871 (119909 119910) 119865(119909 119910)119901minus1]
]
119889119909119889119910
=119901
119902 minus 1
times int
119889
119888
int
119887
119886
[[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
1119901
times log(119867(119877 119877
1015840
)
119867 (119909 119910))119908(119909 119910)
1119901
119867(119909 119910)(119901minus1)119901
times ℎ(119909 119910)minus(119901minus1)119901
119871 (119909 119910)]]
]
times[[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
(119901minus1)119901
times 119908(119909 119910)(119901minus1)119901
times [119867(119909 119910)minus1
times ℎ (119909 119910)](119901minus1)119901
times 119865(119909 119910)119901minus1]
]
]
119889119909119889119910
(32)
By applying Holderrsquos inequality with indices 119901 119901(119901 minus 1) onthe right side of (32) we obtain
int
119889
119888
int
119887
119886
119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119865(119909 119910)119901
119889119909 119889119910
le 120573(119901
119902 minus 1)
times
int
119889
119888
int
119887
119886
[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
119901
119908 (119909 119910)119867(119909 119910)119901minus1
times ℎ(119909 119910)minus(119901minus1)
119871(119909 119910)119901]
]
119889119909119889119910
1119901
times
int
119889
119888
int
119887
119886
[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
times 119908 (119909 119910)119867(119909 119910)minus1
times ℎ (119909 119910) 119865(119909 119910)119901]
]
119889119909119889119910
(119901minus1)119901
(33)
Dividing both sides of (33) by the second integral factor onthe right side of (33) and raising both sides to the 119901th powerwe obtain
int
119889
119888
int
119887
119886
119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
times 119865(119909 119910)119901
119889119909 119889119910
le [120573(119901
119902 minus 1)]
119901
times int
119889
119888
int
119887
119886
[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
119901minus119902
119908 (119909 119910)119867(119909 119910)119901minus1
times ℎ(119909 119910)minus(119901minus1)
119871(119909 119910)119901]
]
119889119909119889119910
(34)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Chang-Jian Zhaorsquos research is supported by National NaturalScience Foundation of China (11371334)Wing-sumCheungrsquosresearch is partially supported by HKU URG grant
References
[1] G H Hardy ldquoNote on a theorem of Hilbertrdquo MathematischeZeitschrift vol 6 no 3-4 pp 314ndash317 1920
[2] G H Hardy J E Littlewood and G Polya InequalitiesCambridge University Press Cambridge UK 1934
[3] G H Hardy ldquoNotes on some points in the integral calculusrdquoMessenger of Mathematics vol 57 pp 12ndash16 1928
The Scientific World Journal 7
[4] B G Pachpatte ldquoOn some generalizations of Hardyrsquos integralinequalityrdquo Journal of Mathematical Analysis and Applicationsvol 234 no 1 pp 15ndash30 1999
[5] B G Pachpatte ldquoOn Hardy type integral inequalities forfunctions of two variablesrdquo Demonstratio Mathematica vol 28no 2 pp 239ndash244 1995
[6] J E Pecaric andE R Love ldquoStillmore generalizations ofHardyrsquosinequalityrdquo Journal of the Australian Mathematical Society vol58 pp 1ndash11 1995
[7] E R Love ldquoGeneralizations of Hardys inequalityrdquo Proceedingsof the Royal Society of Edinburgh vol 100 pp 237ndash262 1985
[8] B C Yang I Brnetic M Krnic and J Pecaric ldquoGeneralizationof Hilbert and Hardy-Hilbert integral inequalitiesrdquo Mathemat-ical Inequalities amp Applications vol 8 pp 259ndash272 2005
[9] B Yang and L Debnath ldquoOn the extended Hardy-Hilbertrsquosinequalityrdquo Journal of Mathematical Analysis and Applicationsvol 272 no 1 pp 187ndash199 2002
[10] K Jichang and L Debnath ldquoOn new generalizations of Hilbertrsquosinequality and their applicationsrdquo Journal of MathematicalAnalysis and Applications vol 245 no 1 pp 248ndash265 2000
[11] M Z Sarikaya and H Yildirim ldquoSome Hardy type integralinequalitiesrdquo Journal of Inequalities in Pure and Applied Mathe-matics vol 7 no 5 article 178 2006
[12] C J Zhao and L Debnath ldquoSome new inverse type Hilbertintegral inequalitiesrdquo Journal of Mathematical Analysis andApplications vol 262 pp 411ndash418 2001
[13] V M Miklyukov and M K Vuorinen ldquoHardyrsquos inequality forw11199010-functions on riemannian manifoldsrdquo Proceedings of the
American Mathematical Society vol 127 no 9 pp 2745ndash27541999
[14] B Opic and A Kufner Hardy-Type Inequalities LongmanEssex UK 1990
[15] A Kufner and L E Persson Weighted Inequalities of HardyType World Scientific 2003
[16] T Leng and Y Feng ldquoOn Hardy-type integral inequalitiesrdquoApplied Mathematics and Mechanics vol 34 no 10 pp 1297ndash1304 2013
[17] E T Copson ldquoSome integral inequalitiesrdquo Proceedings of theRoyal Society of Edinburgh vol 75 pp 157ndash164 1976
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 The Scientific World Journal
From (18) (20) and integrating by parts for 119909 we have
int
119889
119888
int
119887
119886
119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119865(119909 119910)119901
119889119909 119889119910
= minusint
119889
119888
119908(119909 119910) 119865(119909 119910)119901
times[log (119867 (119877 119877
1015840
) 119867 (119909 119910))]minus119902+1
minus119902 + 1
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119909=119887
119909=119886
minus int
119887
119886
[log (119867 (119877 1198771015840
) 119867 (119909 119910))]minus119902+1
minus119902 + 1
times [120597119908 (119909 119910)
120597119909119865(119909 119910)
119901
+ 119908 (119909 119910) 119901119865(119909 119910)119901minus1
times (119866 (119909 119910) minus120597119903 (119909 119910) 120597119909
1199032 (119909 119910)
times int
119909
119886
int
119910
119888
119903 (119904 119905) ℎ (119904 119905)
times119891(119904119905)119889119904 119889119905)]119889119909
119889119910
(21)
where
119866 (119909 119910) =1
119903 (119909 119910)int
119910
119888
119903 (119909 119905) ℎ (119909 119905) 119891 (119909 119905) 119889119905 (22)
If 119902 lt 1 then we observe that
int
119889
119888
int
119887
119886
119863(119909 119910)119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119865(119909 119910)119901
119889119909 119889119910
le119901
1 minus 119902int
119889
119888
int
119887
119886
[
[
119908 (119909 119910)(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902+1
times 119866 (119909 119910) 119865(119909 119910)119901minus1]
]
119889119909119889119910
=119901
1 minus 119902
times int
119889
119888
int
119887
119886
[[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
1119901
times log(119867(119877 119877
1015840
)
119867 (119909 119910))119908(119909 119910)
1119901
times 119867(119909 119910)(119901minus1)119901
times ℎ(119909 119910)minus(119901minus1)119901
119866 (119909 119910)]]
]
times[[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
(119901minus1)119901
times 119908(119909 119910)(119901minus1)119901
times [119867(119909 119910)minus1
times ℎ (119909 119910)](119901minus1)119901
times 119865(119909 119910)119901minus1]
]
]
119889119909119889119910
(23)
By applying Holderrsquos inequality with indices 119901 119901(119901 minus 1) onthe right side of (23) we obtain
int
119889
119888
int
119887
119886
119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119865(119909 119910)119901
119889119909 119889119910
le 120572(119901
1 minus 119902)
times
int
119889
119888
int
119887
119886
[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
119901
times 119908 (119909 119910)119867(119909 119910)119901minus1
times ℎ(119909 119910)minus(119901minus1)
119866(119909 119910)119901]
]
119889119909119889119910
1119901
The Scientific World Journal 5
times
int
119889
119888
int
119887
119886
[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119908 (119909 119910)119867(119909 119910)minus1
times ℎ (119909 119910) 119865(119909 119910)119901]
]
119889119909119889119910
(119901minus1)119901
(24)
Dividing both sides of (24) by the second integral factor onthe right side of (24) and raising both sides to the 119901th powerwe obtain
int
119889
119888
int
119887
119886
119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119865(119909 119910)119901
119889119909 119889119910
le [120572(119901
1 minus 119902)]
119901
times int
119889
119888
int
119887
119886
[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
119901minus119902
times 119908 (119909 119910)119867(119909 119910)119901minus1
times ℎ(119909 119910)minus(119901minus1)
119866(119909 119910)119901]
]
119889119909119889119910
(25)
Proof of Theorem 3 If we let 119906(119909 119910) = 119908(119909 119910)119865(119909 119910)119901 and in
view of
119865 (119909 119910) =1
119903 (119909 119910)int
119887
119909
int
119889
119910
119903 (119904 119905) ℎ (119904 119905) 119891 (119904 119905) 119889119904 119889119905 (26)
for (119909 119910) isin (119886 119887) times (119888 119889) then
120597119906 (119909 119910)
120597119910
=120597119908 (119909 119910)
120597119910119865(119909 119910)
119901
+ 119908 (119909 119910) 119901119865(119909 119910)119901minus1
times (minus1
119903 (119909 119910)int
119889
119910
119903 (119909 119905) ℎ (119909 119905) 119891 (119909 119905) 119889119905
minus120597119903 (119909 119910) 120597119910
1199032 (119909 119910)int
119887
119909
int
119889
119910
119903 (119904 119905) ℎ (119904 119905) 119891 (119904 119905) 119889119904 119889119905)
(27)
Let
120597V (119909 119910)120597119910
= 119867(119909 119910)minus1
ℎ (119909 119910) [log(119867(119877 119877
1015840
)
119867 (119909 119910))]
minus119902
(28)
where ℎ(119909 119910) = int119909
119886
ℎ(119904 119910)119889119904 and in view of 119867(119909 119910) =
int119909
119886
int119910
119888
ℎ(119904 119905)119889119904 119889119905 for (119909 119910) isin (119886 119887) times (119888 119889) then
V (119909 119910) = minus[log (119867 (119877 119877
1015840
) 119867 (119909 119910))]minus119902+1
(minus119902 + 1) (29)
From (27) (29) and integrating by parts for 119910 we have
int
119889
119888
int
119887
119886
119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119865(119909 119910)119901
119889119909 119889119910
= minusint
119887
119886
119908(119909 119910) 119865(119909 119910)119901
times[log (119867 (119877 119877
1015840
) 119867 (119909 119910))]minus119902+1
minus119902 + 1
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119910=119889
119910=119888
minus int
119889
119888
[log (119867 (119877 1198771015840
) 119867 (119909 119910))]minus119902+1
minus119902 + 1
times [120597119908 (119909 119910)
120597119910119865(119909 119910)
119901
+ 119908 (119909 119910) 119901119865(119909 119910)119901minus1
times (119871 (119909 119910) minus120597119903 (119909 119910) 120597119910
1199032 (119909 119910)
times int
119887
119909
int
119889
119910
119903 (119904 119905) ℎ (119904 119905)
times119891(119904 119905)119889119904 119889119905)]119889119910
119889119909
(30)
where
119871 (119909 119910) = minus1
119903 (119909 119910)int
119887
119909
119903 (119904 119910) ℎ (119904 119910) 119891 (119904 119910) 119889119904 (31)
If 119902 gt 1 then we observe that
int
119889
119888
int
119887
119886
119864 (119909 119910)119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119865(119909 119910)119901
119889119909 119889119910
6 The Scientific World Journal
le119901
119902 minus 1int
119889
119888
int
119887
119886
[
[
119908 (119909 119910)(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902+1
times 119871 (119909 119910) 119865(119909 119910)119901minus1]
]
119889119909119889119910
=119901
119902 minus 1
times int
119889
119888
int
119887
119886
[[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
1119901
times log(119867(119877 119877
1015840
)
119867 (119909 119910))119908(119909 119910)
1119901
119867(119909 119910)(119901minus1)119901
times ℎ(119909 119910)minus(119901minus1)119901
119871 (119909 119910)]]
]
times[[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
(119901minus1)119901
times 119908(119909 119910)(119901minus1)119901
times [119867(119909 119910)minus1
times ℎ (119909 119910)](119901minus1)119901
times 119865(119909 119910)119901minus1]
]
]
119889119909119889119910
(32)
By applying Holderrsquos inequality with indices 119901 119901(119901 minus 1) onthe right side of (32) we obtain
int
119889
119888
int
119887
119886
119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119865(119909 119910)119901
119889119909 119889119910
le 120573(119901
119902 minus 1)
times
int
119889
119888
int
119887
119886
[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
119901
119908 (119909 119910)119867(119909 119910)119901minus1
times ℎ(119909 119910)minus(119901minus1)
119871(119909 119910)119901]
]
119889119909119889119910
1119901
times
int
119889
119888
int
119887
119886
[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
times 119908 (119909 119910)119867(119909 119910)minus1
times ℎ (119909 119910) 119865(119909 119910)119901]
]
119889119909119889119910
(119901minus1)119901
(33)
Dividing both sides of (33) by the second integral factor onthe right side of (33) and raising both sides to the 119901th powerwe obtain
int
119889
119888
int
119887
119886
119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
times 119865(119909 119910)119901
119889119909 119889119910
le [120573(119901
119902 minus 1)]
119901
times int
119889
119888
int
119887
119886
[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
119901minus119902
119908 (119909 119910)119867(119909 119910)119901minus1
times ℎ(119909 119910)minus(119901minus1)
119871(119909 119910)119901]
]
119889119909119889119910
(34)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Chang-Jian Zhaorsquos research is supported by National NaturalScience Foundation of China (11371334)Wing-sumCheungrsquosresearch is partially supported by HKU URG grant
References
[1] G H Hardy ldquoNote on a theorem of Hilbertrdquo MathematischeZeitschrift vol 6 no 3-4 pp 314ndash317 1920
[2] G H Hardy J E Littlewood and G Polya InequalitiesCambridge University Press Cambridge UK 1934
[3] G H Hardy ldquoNotes on some points in the integral calculusrdquoMessenger of Mathematics vol 57 pp 12ndash16 1928
The Scientific World Journal 7
[4] B G Pachpatte ldquoOn some generalizations of Hardyrsquos integralinequalityrdquo Journal of Mathematical Analysis and Applicationsvol 234 no 1 pp 15ndash30 1999
[5] B G Pachpatte ldquoOn Hardy type integral inequalities forfunctions of two variablesrdquo Demonstratio Mathematica vol 28no 2 pp 239ndash244 1995
[6] J E Pecaric andE R Love ldquoStillmore generalizations ofHardyrsquosinequalityrdquo Journal of the Australian Mathematical Society vol58 pp 1ndash11 1995
[7] E R Love ldquoGeneralizations of Hardys inequalityrdquo Proceedingsof the Royal Society of Edinburgh vol 100 pp 237ndash262 1985
[8] B C Yang I Brnetic M Krnic and J Pecaric ldquoGeneralizationof Hilbert and Hardy-Hilbert integral inequalitiesrdquo Mathemat-ical Inequalities amp Applications vol 8 pp 259ndash272 2005
[9] B Yang and L Debnath ldquoOn the extended Hardy-Hilbertrsquosinequalityrdquo Journal of Mathematical Analysis and Applicationsvol 272 no 1 pp 187ndash199 2002
[10] K Jichang and L Debnath ldquoOn new generalizations of Hilbertrsquosinequality and their applicationsrdquo Journal of MathematicalAnalysis and Applications vol 245 no 1 pp 248ndash265 2000
[11] M Z Sarikaya and H Yildirim ldquoSome Hardy type integralinequalitiesrdquo Journal of Inequalities in Pure and Applied Mathe-matics vol 7 no 5 article 178 2006
[12] C J Zhao and L Debnath ldquoSome new inverse type Hilbertintegral inequalitiesrdquo Journal of Mathematical Analysis andApplications vol 262 pp 411ndash418 2001
[13] V M Miklyukov and M K Vuorinen ldquoHardyrsquos inequality forw11199010-functions on riemannian manifoldsrdquo Proceedings of the
American Mathematical Society vol 127 no 9 pp 2745ndash27541999
[14] B Opic and A Kufner Hardy-Type Inequalities LongmanEssex UK 1990
[15] A Kufner and L E Persson Weighted Inequalities of HardyType World Scientific 2003
[16] T Leng and Y Feng ldquoOn Hardy-type integral inequalitiesrdquoApplied Mathematics and Mechanics vol 34 no 10 pp 1297ndash1304 2013
[17] E T Copson ldquoSome integral inequalitiesrdquo Proceedings of theRoyal Society of Edinburgh vol 75 pp 157ndash164 1976
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
The Scientific World Journal 5
times
int
119889
119888
int
119887
119886
[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119908 (119909 119910)119867(119909 119910)minus1
times ℎ (119909 119910) 119865(119909 119910)119901]
]
119889119909119889119910
(119901minus1)119901
(24)
Dividing both sides of (24) by the second integral factor onthe right side of (24) and raising both sides to the 119901th powerwe obtain
int
119889
119888
int
119887
119886
119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119865(119909 119910)119901
119889119909 119889119910
le [120572(119901
1 minus 119902)]
119901
times int
119889
119888
int
119887
119886
[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
119901minus119902
times 119908 (119909 119910)119867(119909 119910)119901minus1
times ℎ(119909 119910)minus(119901minus1)
119866(119909 119910)119901]
]
119889119909119889119910
(25)
Proof of Theorem 3 If we let 119906(119909 119910) = 119908(119909 119910)119865(119909 119910)119901 and in
view of
119865 (119909 119910) =1
119903 (119909 119910)int
119887
119909
int
119889
119910
119903 (119904 119905) ℎ (119904 119905) 119891 (119904 119905) 119889119904 119889119905 (26)
for (119909 119910) isin (119886 119887) times (119888 119889) then
120597119906 (119909 119910)
120597119910
=120597119908 (119909 119910)
120597119910119865(119909 119910)
119901
+ 119908 (119909 119910) 119901119865(119909 119910)119901minus1
times (minus1
119903 (119909 119910)int
119889
119910
119903 (119909 119905) ℎ (119909 119905) 119891 (119909 119905) 119889119905
minus120597119903 (119909 119910) 120597119910
1199032 (119909 119910)int
119887
119909
int
119889
119910
119903 (119904 119905) ℎ (119904 119905) 119891 (119904 119905) 119889119904 119889119905)
(27)
Let
120597V (119909 119910)120597119910
= 119867(119909 119910)minus1
ℎ (119909 119910) [log(119867(119877 119877
1015840
)
119867 (119909 119910))]
minus119902
(28)
where ℎ(119909 119910) = int119909
119886
ℎ(119904 119910)119889119904 and in view of 119867(119909 119910) =
int119909
119886
int119910
119888
ℎ(119904 119905)119889119904 119889119905 for (119909 119910) isin (119886 119887) times (119888 119889) then
V (119909 119910) = minus[log (119867 (119877 119877
1015840
) 119867 (119909 119910))]minus119902+1
(minus119902 + 1) (29)
From (27) (29) and integrating by parts for 119910 we have
int
119889
119888
int
119887
119886
119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119865(119909 119910)119901
119889119909 119889119910
= minusint
119887
119886
119908(119909 119910) 119865(119909 119910)119901
times[log (119867 (119877 119877
1015840
) 119867 (119909 119910))]minus119902+1
minus119902 + 1
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119910=119889
119910=119888
minus int
119889
119888
[log (119867 (119877 1198771015840
) 119867 (119909 119910))]minus119902+1
minus119902 + 1
times [120597119908 (119909 119910)
120597119910119865(119909 119910)
119901
+ 119908 (119909 119910) 119901119865(119909 119910)119901minus1
times (119871 (119909 119910) minus120597119903 (119909 119910) 120597119910
1199032 (119909 119910)
times int
119887
119909
int
119889
119910
119903 (119904 119905) ℎ (119904 119905)
times119891(119904 119905)119889119904 119889119905)]119889119910
119889119909
(30)
where
119871 (119909 119910) = minus1
119903 (119909 119910)int
119887
119909
119903 (119904 119910) ℎ (119904 119910) 119891 (119904 119910) 119889119904 (31)
If 119902 gt 1 then we observe that
int
119889
119888
int
119887
119886
119864 (119909 119910)119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119865(119909 119910)119901
119889119909 119889119910
6 The Scientific World Journal
le119901
119902 minus 1int
119889
119888
int
119887
119886
[
[
119908 (119909 119910)(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902+1
times 119871 (119909 119910) 119865(119909 119910)119901minus1]
]
119889119909119889119910
=119901
119902 minus 1
times int
119889
119888
int
119887
119886
[[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
1119901
times log(119867(119877 119877
1015840
)
119867 (119909 119910))119908(119909 119910)
1119901
119867(119909 119910)(119901minus1)119901
times ℎ(119909 119910)minus(119901minus1)119901
119871 (119909 119910)]]
]
times[[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
(119901minus1)119901
times 119908(119909 119910)(119901minus1)119901
times [119867(119909 119910)minus1
times ℎ (119909 119910)](119901minus1)119901
times 119865(119909 119910)119901minus1]
]
]
119889119909119889119910
(32)
By applying Holderrsquos inequality with indices 119901 119901(119901 minus 1) onthe right side of (32) we obtain
int
119889
119888
int
119887
119886
119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119865(119909 119910)119901
119889119909 119889119910
le 120573(119901
119902 minus 1)
times
int
119889
119888
int
119887
119886
[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
119901
119908 (119909 119910)119867(119909 119910)119901minus1
times ℎ(119909 119910)minus(119901minus1)
119871(119909 119910)119901]
]
119889119909119889119910
1119901
times
int
119889
119888
int
119887
119886
[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
times 119908 (119909 119910)119867(119909 119910)minus1
times ℎ (119909 119910) 119865(119909 119910)119901]
]
119889119909119889119910
(119901minus1)119901
(33)
Dividing both sides of (33) by the second integral factor onthe right side of (33) and raising both sides to the 119901th powerwe obtain
int
119889
119888
int
119887
119886
119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
times 119865(119909 119910)119901
119889119909 119889119910
le [120573(119901
119902 minus 1)]
119901
times int
119889
119888
int
119887
119886
[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
119901minus119902
119908 (119909 119910)119867(119909 119910)119901minus1
times ℎ(119909 119910)minus(119901minus1)
119871(119909 119910)119901]
]
119889119909119889119910
(34)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Chang-Jian Zhaorsquos research is supported by National NaturalScience Foundation of China (11371334)Wing-sumCheungrsquosresearch is partially supported by HKU URG grant
References
[1] G H Hardy ldquoNote on a theorem of Hilbertrdquo MathematischeZeitschrift vol 6 no 3-4 pp 314ndash317 1920
[2] G H Hardy J E Littlewood and G Polya InequalitiesCambridge University Press Cambridge UK 1934
[3] G H Hardy ldquoNotes on some points in the integral calculusrdquoMessenger of Mathematics vol 57 pp 12ndash16 1928
The Scientific World Journal 7
[4] B G Pachpatte ldquoOn some generalizations of Hardyrsquos integralinequalityrdquo Journal of Mathematical Analysis and Applicationsvol 234 no 1 pp 15ndash30 1999
[5] B G Pachpatte ldquoOn Hardy type integral inequalities forfunctions of two variablesrdquo Demonstratio Mathematica vol 28no 2 pp 239ndash244 1995
[6] J E Pecaric andE R Love ldquoStillmore generalizations ofHardyrsquosinequalityrdquo Journal of the Australian Mathematical Society vol58 pp 1ndash11 1995
[7] E R Love ldquoGeneralizations of Hardys inequalityrdquo Proceedingsof the Royal Society of Edinburgh vol 100 pp 237ndash262 1985
[8] B C Yang I Brnetic M Krnic and J Pecaric ldquoGeneralizationof Hilbert and Hardy-Hilbert integral inequalitiesrdquo Mathemat-ical Inequalities amp Applications vol 8 pp 259ndash272 2005
[9] B Yang and L Debnath ldquoOn the extended Hardy-Hilbertrsquosinequalityrdquo Journal of Mathematical Analysis and Applicationsvol 272 no 1 pp 187ndash199 2002
[10] K Jichang and L Debnath ldquoOn new generalizations of Hilbertrsquosinequality and their applicationsrdquo Journal of MathematicalAnalysis and Applications vol 245 no 1 pp 248ndash265 2000
[11] M Z Sarikaya and H Yildirim ldquoSome Hardy type integralinequalitiesrdquo Journal of Inequalities in Pure and Applied Mathe-matics vol 7 no 5 article 178 2006
[12] C J Zhao and L Debnath ldquoSome new inverse type Hilbertintegral inequalitiesrdquo Journal of Mathematical Analysis andApplications vol 262 pp 411ndash418 2001
[13] V M Miklyukov and M K Vuorinen ldquoHardyrsquos inequality forw11199010-functions on riemannian manifoldsrdquo Proceedings of the
American Mathematical Society vol 127 no 9 pp 2745ndash27541999
[14] B Opic and A Kufner Hardy-Type Inequalities LongmanEssex UK 1990
[15] A Kufner and L E Persson Weighted Inequalities of HardyType World Scientific 2003
[16] T Leng and Y Feng ldquoOn Hardy-type integral inequalitiesrdquoApplied Mathematics and Mechanics vol 34 no 10 pp 1297ndash1304 2013
[17] E T Copson ldquoSome integral inequalitiesrdquo Proceedings of theRoyal Society of Edinburgh vol 75 pp 157ndash164 1976
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 The Scientific World Journal
le119901
119902 minus 1int
119889
119888
int
119887
119886
[
[
119908 (119909 119910)(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902+1
times 119871 (119909 119910) 119865(119909 119910)119901minus1]
]
119889119909119889119910
=119901
119902 minus 1
times int
119889
119888
int
119887
119886
[[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
1119901
times log(119867(119877 119877
1015840
)
119867 (119909 119910))119908(119909 119910)
1119901
119867(119909 119910)(119901minus1)119901
times ℎ(119909 119910)minus(119901minus1)119901
119871 (119909 119910)]]
]
times[[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
(119901minus1)119901
times 119908(119909 119910)(119901minus1)119901
times [119867(119909 119910)minus1
times ℎ (119909 119910)](119901minus1)119901
times 119865(119909 119910)119901minus1]
]
]
119889119909119889119910
(32)
By applying Holderrsquos inequality with indices 119901 119901(119901 minus 1) onthe right side of (32) we obtain
int
119889
119888
int
119887
119886
119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
119865(119909 119910)119901
119889119909 119889119910
le 120573(119901
119902 minus 1)
times
int
119889
119888
int
119887
119886
[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
119901
119908 (119909 119910)119867(119909 119910)119901minus1
times ℎ(119909 119910)minus(119901minus1)
119871(119909 119910)119901]
]
119889119909119889119910
1119901
times
int
119889
119888
int
119887
119886
[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
times 119908 (119909 119910)119867(119909 119910)minus1
times ℎ (119909 119910) 119865(119909 119910)119901]
]
119889119909119889119910
(119901minus1)119901
(33)
Dividing both sides of (33) by the second integral factor onthe right side of (33) and raising both sides to the 119901th powerwe obtain
int
119889
119888
int
119887
119886
119908 (119909 119910)119867(119909 119910)minus1
ℎ (119909 119910)
times (log(119867(119877 119877
1015840
)
119867 (119909 119910)))
minus119902
times 119865(119909 119910)119901
119889119909 119889119910
le [120573(119901
119902 minus 1)]
119901
times int
119889
119888
int
119887
119886
[
[
(log(119867(119877 119877
1015840
)
119867 (119909 119910)))
119901minus119902
119908 (119909 119910)119867(119909 119910)119901minus1
times ℎ(119909 119910)minus(119901minus1)
119871(119909 119910)119901]
]
119889119909119889119910
(34)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Chang-Jian Zhaorsquos research is supported by National NaturalScience Foundation of China (11371334)Wing-sumCheungrsquosresearch is partially supported by HKU URG grant
References
[1] G H Hardy ldquoNote on a theorem of Hilbertrdquo MathematischeZeitschrift vol 6 no 3-4 pp 314ndash317 1920
[2] G H Hardy J E Littlewood and G Polya InequalitiesCambridge University Press Cambridge UK 1934
[3] G H Hardy ldquoNotes on some points in the integral calculusrdquoMessenger of Mathematics vol 57 pp 12ndash16 1928
The Scientific World Journal 7
[4] B G Pachpatte ldquoOn some generalizations of Hardyrsquos integralinequalityrdquo Journal of Mathematical Analysis and Applicationsvol 234 no 1 pp 15ndash30 1999
[5] B G Pachpatte ldquoOn Hardy type integral inequalities forfunctions of two variablesrdquo Demonstratio Mathematica vol 28no 2 pp 239ndash244 1995
[6] J E Pecaric andE R Love ldquoStillmore generalizations ofHardyrsquosinequalityrdquo Journal of the Australian Mathematical Society vol58 pp 1ndash11 1995
[7] E R Love ldquoGeneralizations of Hardys inequalityrdquo Proceedingsof the Royal Society of Edinburgh vol 100 pp 237ndash262 1985
[8] B C Yang I Brnetic M Krnic and J Pecaric ldquoGeneralizationof Hilbert and Hardy-Hilbert integral inequalitiesrdquo Mathemat-ical Inequalities amp Applications vol 8 pp 259ndash272 2005
[9] B Yang and L Debnath ldquoOn the extended Hardy-Hilbertrsquosinequalityrdquo Journal of Mathematical Analysis and Applicationsvol 272 no 1 pp 187ndash199 2002
[10] K Jichang and L Debnath ldquoOn new generalizations of Hilbertrsquosinequality and their applicationsrdquo Journal of MathematicalAnalysis and Applications vol 245 no 1 pp 248ndash265 2000
[11] M Z Sarikaya and H Yildirim ldquoSome Hardy type integralinequalitiesrdquo Journal of Inequalities in Pure and Applied Mathe-matics vol 7 no 5 article 178 2006
[12] C J Zhao and L Debnath ldquoSome new inverse type Hilbertintegral inequalitiesrdquo Journal of Mathematical Analysis andApplications vol 262 pp 411ndash418 2001
[13] V M Miklyukov and M K Vuorinen ldquoHardyrsquos inequality forw11199010-functions on riemannian manifoldsrdquo Proceedings of the
American Mathematical Society vol 127 no 9 pp 2745ndash27541999
[14] B Opic and A Kufner Hardy-Type Inequalities LongmanEssex UK 1990
[15] A Kufner and L E Persson Weighted Inequalities of HardyType World Scientific 2003
[16] T Leng and Y Feng ldquoOn Hardy-type integral inequalitiesrdquoApplied Mathematics and Mechanics vol 34 no 10 pp 1297ndash1304 2013
[17] E T Copson ldquoSome integral inequalitiesrdquo Proceedings of theRoyal Society of Edinburgh vol 75 pp 157ndash164 1976
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
The Scientific World Journal 7
[4] B G Pachpatte ldquoOn some generalizations of Hardyrsquos integralinequalityrdquo Journal of Mathematical Analysis and Applicationsvol 234 no 1 pp 15ndash30 1999
[5] B G Pachpatte ldquoOn Hardy type integral inequalities forfunctions of two variablesrdquo Demonstratio Mathematica vol 28no 2 pp 239ndash244 1995
[6] J E Pecaric andE R Love ldquoStillmore generalizations ofHardyrsquosinequalityrdquo Journal of the Australian Mathematical Society vol58 pp 1ndash11 1995
[7] E R Love ldquoGeneralizations of Hardys inequalityrdquo Proceedingsof the Royal Society of Edinburgh vol 100 pp 237ndash262 1985
[8] B C Yang I Brnetic M Krnic and J Pecaric ldquoGeneralizationof Hilbert and Hardy-Hilbert integral inequalitiesrdquo Mathemat-ical Inequalities amp Applications vol 8 pp 259ndash272 2005
[9] B Yang and L Debnath ldquoOn the extended Hardy-Hilbertrsquosinequalityrdquo Journal of Mathematical Analysis and Applicationsvol 272 no 1 pp 187ndash199 2002
[10] K Jichang and L Debnath ldquoOn new generalizations of Hilbertrsquosinequality and their applicationsrdquo Journal of MathematicalAnalysis and Applications vol 245 no 1 pp 248ndash265 2000
[11] M Z Sarikaya and H Yildirim ldquoSome Hardy type integralinequalitiesrdquo Journal of Inequalities in Pure and Applied Mathe-matics vol 7 no 5 article 178 2006
[12] C J Zhao and L Debnath ldquoSome new inverse type Hilbertintegral inequalitiesrdquo Journal of Mathematical Analysis andApplications vol 262 pp 411ndash418 2001
[13] V M Miklyukov and M K Vuorinen ldquoHardyrsquos inequality forw11199010-functions on riemannian manifoldsrdquo Proceedings of the
American Mathematical Society vol 127 no 9 pp 2745ndash27541999
[14] B Opic and A Kufner Hardy-Type Inequalities LongmanEssex UK 1990
[15] A Kufner and L E Persson Weighted Inequalities of HardyType World Scientific 2003
[16] T Leng and Y Feng ldquoOn Hardy-type integral inequalitiesrdquoApplied Mathematics and Mechanics vol 34 no 10 pp 1297ndash1304 2013
[17] E T Copson ldquoSome integral inequalitiesrdquo Proceedings of theRoyal Society of Edinburgh vol 75 pp 157ndash164 1976
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
