Miskolc Mathematical Notes HU e-ISSN 1787-2413Vol. 20 (2019), No. 1, pp. 451–463 DOI: 10.18514/MMN.2019.2608

ON .p;q/–EXTENSION OF FURTHER MEMBERS OFBESSEL–STRUVE FUNCTIONS CLASS

RAKESH K. PARMAR AND TIBOR K. POGANY

Received 23 April, 2018

Abstract. In [10] .p;q/–extensions of the modified Bessel and the modified Struve functionsof the first kind are presented. This article companion to [10] contains the .p;q/–extensionof modified Struve function of the second kind M�;p;q and the Bessel-Struve kernel functionS�;p;q . Systematic investigation of its properties, among integral representation, Mellin trans-form, Laguerre polynomial representation for both introduced special functions, while additionaldifferential–difference equation, log–convexity property and Turan–type inequalities are realizedfor the latter.

2010 Mathematics Subject Classification: Primary 33C10; 33E20; Secondary 26D15; 33C45

Keywords: .p;q/-extended Beta function, .p;q/-extended Bessel and modified Bessel and Struvefunctions of the first and second kind, .p;q/–extended Bessel-Struve kernel function, integralrepresentation, log–convexity, Turan–type inequality

1. INTRODUCTION AND PRELIMINARIES

The theory of special functions has been one of the most rapidly growing researchsubjects in Mathematical Analysis due to their diverse applications in many differentareas of mathematical, physical, statistical, and engineering sciences. Further, ex-tensions of the Eulerian Gamma function enable to define important generalizationsnot only of the higher transcendental hypereoetric functions, but the Bessel and alikefunctions class members. So, bearing in mind that

4�n

nŠ � .nC�C1/D

B�nC 1

2;�C 1

2

�p� �

��C 1

2

�.2n/Š

; 2� > �1; n 2N0 ;

the Bessel function of the first kind can be rewritten into

J�.´/DXn�0

.�1/n�´2

�2nC�nŠ� .nC�C1/

D

�´2

��p� � .�C 1

2/

Xn�0

B�nC 1

2; �C 1

2

� .�´2/n.2n/Š

:

Then making use of the .p;q/–extended Beta function [7] ( see also, [18])

B.x;yIp;q/DZ 1

0

tx�1.1� t /y�1 e�pt�

q1�t dt ;

c 2019 Miskolc University Press

452 RAKESH K. PARMAR AND TIBOR K. POGANY

when minf<.x/;<.y/g > 0Iminf<.p/;<.q/g � 0 to J�.´/ several extensions, gen-eralizations and unifications of various special functions of .p;q/–variant, and in turnthe .p;p/–, that is the p–variant have been studied widely together with the set of re-lated higher transcendental hypergeometric type special functions by several authors,consult for instance [3–6, 12, 15, 18]). In particular, Masirevic et al. [10] applied the.p;q/–extended Beta in introducing the .p;q/–Bessel function of the first kind

J�;p;q.´/D

�´2

��p� � .�C 1

2/

Xn�0

B�nC

1

2; �C

1

2Ip;q

�.�´2/n

.2n/Š: (1.1)

These publication also contains the newly defined the .p;q/–extended modified BesselI�;p;q together with the modified Struve L�;p;q functions of the first kind of order �in the form:

I�;p;q.´/D

�´2

��p� � .�C 1

2/

Xn�0

B�nC

1

2; �C

1

2Ip;q

�´2n

.2n/Š

L�;p;q.´/D2�´2

��C1p� � .�C 1

2/

Xn�0

B�nC1; �C

1

2Ip;q

�´2n

.2nC1/Š;

where minfp;qg � 0 while for p D 0 D q necessarily <.�/ > �12

. Among otherfindings they reported about the integral representations [10]

I�;p;q.´/D2�´2

��p� � .�C 1

2/

Z 1

0

.1� t2/��12 cosh.´t/e�

p

t2�q

1�t2 dt; (1.2)

L�;p;q.´/D2�´2

��p� � .�C 1

2/

Z 1

0

.1� t2/��12 sinh.´t/e�

p

t2�q

1�t2 dt: (1.3)

Clearly, for p D 0D q all these extensions reduce to their classical ancestors.Here our aim is to introduce and investigate, in a rather systematic manner, .p;q/–

extended modified Struve function M�;p;q.x/ and the so–called .p;q/–extendedBessel–Struve kernel function S�;p;q.x/ presenting their various analytical proper-ties.

2. THE .p;q/–EXTENDED MODIFIED STRUVE FUNCTION M�;p;q.x/

The series definition of the Fox-Wright function pq.´/ [17] reads

pq

".˛1;A1/; � � � ; . p;Ap/

.ˇ1;B1/; � � � ; .ˇq;Bq/

ˇˇ ´#D

Xn�0

pQjD1

� . j CAjn/

qQjD1

� .ˇkCBkn/

´n

nŠ; (2.1)

.p;q/–EXTENDED BESSEL–STRUVE TYPE FUNCTIONS 453

where Aj ;Bk > 0 j D 1; � � � ;pI k D 1; � � � ;q and the series converges for all ´ 2 Cwhen �D 1C

Pq

kD1Bk �

PpjD1Aj > 0; in the case �D 0 the convergence occurs

inside the open disc

j´j< r D

0@ pYjD1

A�Aj

j

1A0@ qYjD1

BBj

j

1A :The modifed Struve functions of the second kind [2, p. 1388, Eq. (2)], [13, Eq.11.2.6]

M�.x/D L�.x/�I�.x/ ;having integral form [13, Eq. 11.5.4]

M�.x/D�2�x2

��p� � .�C 1

2/

Z 1

0

.1� t2/��12 e�xt dt; 2�C1 > 0;x > 0:

By routine calculation we infer the related power series representation:

Theorem 1. For all 2�C1 > 0;x > 0 we have

M�.x/D�2�� x�

� .�C 12/11

".12; 12/

.�C1; 12/

ˇˇ �x

#:

The proof is straightforward mentioning only that being �D 1, the series convergesfor all x > 0.

We now introduce the .p;q/–extended modified Struve function of the second kindas

M�;p;q.x/D L�;p;q.x/�I�;p;q.x/ :Applying this definition by taking (1.2) and (1.3), we obtain the related integral rep-resentation

M�;p;q.x/D�2�x2

��p� � .�C 1

2/

Z 1

0

.1� t2/��12 e�xt�

p

t2�q

1�t2 dt ; (2.2)

with minfp;qg� 0 and<.�/>�12

when pD qD 0. Clearly, the latter case coincideswith the original definition of modified Struve function M�.x/ of second kind.

The double Mellin transforms [14, p. 293, Eq. (7.1.6)] of a suitable class ofL1.R2C/–integrable function f .x;y/ with respect to the indices r and s is defined by

M ff .x;y/g.r; s/D

Z 10

Z 10

xr�1ys�1 f .x;y/ dx dy; (2.3)

provided that the improper integral in (2.3) exists. Now, we examine the existenceand the establish the shape of the double Mellin transform of M�;p;q.x/ when thearguments are the extension parameters p;q.

454 RAKESH K. PARMAR AND TIBOR K. POGANY

Theorem 2. For all minf<.r/; <.s/; 2<.�/C1g> 0 and x > 0 the Mellin trans-form of M�;p;q.x/ with respect to p;q � 0 reads as follows

M fM�;p;q.x/g.r; s/D�

�x2

��� .r/� .s/� .�C sC 1

2/

p� � .�C 1

2/

� 11

".12C r; 1

2/

.�C rC sC1; 12/

ˇˇ�x

#:

Proof. By the definition of Mellin transform we find from (2.2) that

M˚M�;p;q.x/

.r; s/D

Z 10

Z 10

pr�1qs�1M�;p;q.x/dp dq

D�2�x2

��p� � .�C 1

2/

Z 1

0

.1� t2/��12 e�xt

�

�Z 10

pr�1e�p

t2 dp��Z 1

0

qs�1e�q

1�t2 dq�

dt;

where we have interchanged the integration order. Applying the well-known Gammafunction formula [13, p.136, Eq. (5.2.1)]

� .�/��� D

Z 10

e��t t��1 dt; <.�/ > 0; <.�/ > 0

to the inner p- and q-integrals we get

M˚M�;p;q.x/

.r; s/D�

2�x2

��� .r/� .s/

p� � .�C 1

2/

Z 1

0

t2r .1� t2/�Cs�12 e�xtdt

D�2�x2

��� .r/� .s/

p� � .�C 1

2/

Xn�0

.�x/n

nŠ

Z 1

0

t2rCn .1� t2/�Cs�12 dt

D�

�x2

��� .r/� .s/

p� � .�C 1

2/

Xn�0

.�x/n

nŠ

Z 1

0

urCn2� 1

2 .1�u/�Cs�12 du

D�

�x2

��� .r/� .s/

p� � .�C 1

2/

Xn�0

� .rC n2C12/� .�C sC 1

2/

� .�C rC sC1C n2/

.�x/n

nŠ;

taking into account the series e�xt . Thus, (2.1) finishes the proof. �

Remark 1. If we set r D 1D s in Theorem 2 we getZ 10

Z 10

M�;p;q.x/ dp dq D�

�x2

��.2�C1/

2p�

11

".32; 12/

.�C3; 12/

ˇˇ�x

#; x > 0:

.p;q/–EXTENDED BESSEL–STRUVE TYPE FUNCTIONS 455

Next, we report on the Laguerre polynomial representation for M�;p;q.x/. TheLaguerre polynomial L.˛/n .p/ of the degree n 2N0 is given by the generating func-tion [16, p. 202]

1

.1� t /˛C1exp

��p t

1� t

�D

Xn�0

L.˛/n .p/ tn; ˛ 2CI jt j< 1:

Theorem 3. For all minf<.p/; <.q/g > 0, <.�/ > �12

and ˛ 2 C, for which�C˛ > �3

2, the following Laguerre polynomial representation holds true

M�;p;q.x/D�.�C 1

2/˛C1x

�

p� 2� epCq

Xm;n�0

.�C˛C 32/nL

˛m.p/ L

˛n.q/

� 11

".˛CnC 3

2; 12/

.�C2˛CmCnC3; 12/

ˇˇ�x

#:

Proof. Simple transformation gives

exp��p

t2

�Dt2˛C2

epXm�0

L.˛/m .p/.1� t2/mI

exp��

q

1� t2

�D.1� t2/˛C1

eqXn�0

L.˛/n .q/ t2n ;

that is

e�p

t2�q

1�t2 D e�p�q t2.1� t2/˛C1Xm;n�0

L.˛/m .p/L.˛/n .q/ t2n .1� t2/m : (2.4)

By inserting (2.4) into (2.2) and changing the order of integration and summation, weconclude

M�;p;q.x/D�2�x2

�� e�p�qp� � .�C 1

2/

Xm;n�0

L˛m.p/ L˛n.q/

�

Z 1

0

t2.˛CnC1/ .1� t2/�C˛CmC12 e�xtdt;

and then series expansion of e�xt is used. The rest is obvious. �

We close this section with a certain differential–difference equation for real argu-ment M�;p;q.x/.

Theorem 4. For all minf<.p/;<.q/g > 0 and for <.�/ > �12

when p D q D 0,we have

x2w00� �2� xw0� � Œx

2��.�C1/�w� D�.2�C1/xw�C1; x > 0; (2.5)

where w� D w�.x/�M�;p;q.x/.

456 RAKESH K. PARMAR AND TIBOR K. POGANY

Proof. Upon differentiating both sides of (2.2) with respect to x, we find that�d

dx

�2.x��M�;p;q.x//D�

21��p� � .�C 1

2/

Z 1

0

t2 .1� t2/��12 e�xt�

p

t2�q

1�t2 dt

D�21��

p� � .�C 1

2/

Z 1

0

f1� .1� t2/g.1� t2/��12 e�xt�

p

t2�q

1�t2 dt ;

which can be rewritten as follows:�d

dx

�2.x��M�;p;q.x//D�

21��p� � .�C 1

2/

Z 1

0

.1� t2/��12 e�xt�

p

t2�q

1�t2 dt

C21��

p� � .�C 1

2/

Z 1

0

.1� t2/�C1 e�xt�p

t2�q

1�t2 dt

D1

x�M�;p;q.x/�

2�C1

x�C1M�C1;p;q.x/;

where we use the definition (2.2). Expanding the left-hand-side second derivative andreducing the material we arrive at (2.5). �

3. THE .p;q/–BESSEL–STRUVE KERNEL FUNCTION

The so-called Bessel-Struve kernel function S� is defined by the series [1, p. 1845]

S�.x/D� .�C1/p�

Xn�0

��nC12

���n2C�C1

� xnnŠ; � > �1; x 2 R:

Consider the initial value problem (see [8, 9, 11]):

L� u.x/D �2u.x/; u.0/D 1; u0.0/D

�� .�C1/p� �

��C 3

2

� ;where for � > �1

2, L� stands for the Bessel-Struve operator defined by

L� u.x/Dd2u

dx2.x/C

2�C1

x

�du

dx.x/�

du

dx.0/

�with some u 2 C1.R/. The Bessel–Struve kernel function becomes the particularcase of the unique solution S�.�x/ of the above initial value problem.

The function S� is expressible in terms of the Beta function viz.

S�.x/D� .�C1/p� � .�C 1

2/

Xn�0

B�n

2C1

2; �C

1

2

�xn

nŠ;

and also as a specific Fox–Wright function

S�.x/D 11

".12; 12/

.�C1; 12/

ˇˇ x#; x 2 R;

.p;q/–EXTENDED BESSEL–STRUVE TYPE FUNCTIONS 457

mentioning that S�.0/D 1 and �D 1 which ensures the convergence of this serieson the whole real x axis.

Here, our aim is introducing and studying the novel .p;q/–extended Bessel-Struvekernel function S�;p;q.x/ by presenting functional bound, integral representation,Mellin transform and other properties.

Analogously to (1.1), we introduce the .p;q/–extended Bessel-Struve kernel func-tion S�;p;q.x/ in the form

S�;p;q.x/D� .�C1/p� � .�C 1

2/

Xn�0

B�n

2C1

2; �C

1

2Ip;q

�xn

nŠ: (3.1)

with minfp;qg � 0 and <.�/ > �12

when p D q D 0.Firstly, we derive an integral representation and a functional upper bound for

S�;p;q.x/.

Theorem 5. For all minf<.p/;<.q/g > 0 and for <.�/ > �12

when p D q D 0,we have

S�;p;q.x/D2� .�C1/p� � .�C 1

2/

Z 1

0

.1� t2/��12 ext�

p

t2�q

1�t2 dt; x � 0: (3.2)

Proof. By virtue of the integral expression [10]

B.x;yIp;q/D 2Z 1

0

t2x�1.1� t2/y�1 e�p

t2�q

1�t2 dt ;

valid for all minf<.x/;<.y/g> 0Iminf<.p/;<.q/g � 0, from (3.1) one yields

S�;p;q.x/D2� .�C1/p� � .�C 1

2/

Xn�0

xn

nŠ

Z 1

0

tn .1� t2/��12 e�

p

t2�q

1�t2 dt

D2� .�C1/p� � .�C 1

2/

Z 1

0

.1� t2/��12 e�

p

t2�q

1�t2

Xn�0

.xt/n

nŠdt;

with interchanged order of summation and integration. This is equivalent to the as-sertion. �

Now, making use of the estimate [15, Lemma 2]

sup0<t<1

e�p

t2�q

1�t2 D e�.ppCpq/2 ; minfp;qg � 0;

in (3.2) we deduce

Corollary 1. For all minfp;qg> 0 and for � > �12

if p D q D 0 we have

S�;p;q.x/� e�.ppCpq/2 S�.x/ ; x � 0:

458 RAKESH K. PARMAR AND TIBOR K. POGANY

Theorem 6. For all minf<.p/;<.q/g > 0, for <.�/ > �12

if p D q D 0 and forall x 2 Rn f0g we have

S�;p;q.x/D � .�C1/

�2

x

�� �J�;p;q.ix/� iH�;p;q.ix/

�;

S�;p;q.x/D � .�C1/

�2

x

�� �I�;p;q.x/CL�;p;q.x/

�:

Proof. Using the known integral representations [10]

J�;p;q.x/D2�x2

��p� � .�C 1

2/

Z 1

0

.1� t2/��12 cos.xt/e�

p

t2�q

1�t2 dt;

H�;p;q.x/D2�x2

��p� � .�C 1

2/

Z 1

0

.1� t2/��12 sin.xt/e�

p

t2�q

1�t2 dt;

in right-hand side of the first stated relation we recover S�;p;q.x/. Similarly, with theaid of integral forms [10]

I�;p;q.x/D2�x2

��p� � .�C 1

2/

Z 1

0

.1� t2/��12 cosh.xt/e�

p

t2�q

1�t2 dt;

L�;p;q.x/D2�x2

��p� � .�C 1

2/

Z 1

0

.1� t2/��12 sinh.xt/e�

p

t2�q

1�t2 dt ;

we finish the proof of the second formula. �

Our next goal is the Mellin transform result.

Theorem 7. For all minf<.r/; <.s/g > 0 and <.�/ > �12

the Mellin transformsof S�;p;q with respect to p;q � 0, read as follows:

M fS�;p;q.x/g.r; s/D� .�C1/� .r/� .s/� .�C sC 1

2/

p� � .�C 1

2/

11

h.12C r; 1

2/

.�C rC sC1; 12/

ˇxi:

Proof. We find from (3.2) ad definitionem

M˚S�;p;q.x/

.r; s/D

Z 10

Z 10

pr�1qs�1S�;p;q.x/dp dq

D

Z 10

Z 10

0@� .�C1/pr�1qs�1p� � .�C 1

2/

Xn�0

xnB�n2C12; �C 1

2Ip;q

�nŠ

1Adp dq

D� .�C1/p� � .�C 1

2/

Xn�0

xn

nŠ

Z 10

Z 10

pr�1qs�1B�n2C12; �C 1

2Ip;q

�dp dq:

.p;q/–EXTENDED BESSEL–STRUVE TYPE FUNCTIONS 459

By virtue of the formula [7, p.342, Eq. (2.1)]Z 10

Z 10

pr�1qs�1B.x; y I p;q/dp dq D � .r/� .s/B.xC r; yC s/;

valid for all min˚<.r/; <.s/

> 0, the double integral becomes

M˚S�;p;q.x/

.r; s/D

� .�C1/� .r/� .s/p� � .�C 1

2/

Xn�0

B�n2C12C r; �C 1

2C s

�xn

nŠ:

After simplification we infer the asserted formula. �

Remark 2. For r D 1D s in Theorem 7 we obtain the integration formulaZ 10

Z 10

S�;p;q.x/ dp dq D� .�C1/.�C 1

2/

p�

11

h.32; 12/

.�C3; 12/

ˇxi:

The following Laguerre polynomial representation for S�;p;q.x/ has been estab-lished by the lines of the same fashion result for M�;p;q.x/ in Theorem 3, thereforewe omit the proof.

Theorem 8. For all minf<.p/; <.q/g> 0 and <.�/ > �12

we have

S�;p;q.x/D� .�C1/.�C 1

2/˛C1

p� epCq

Xm;n�0

L˛m.p/ L˛n.q/.�C˛C

32/m

� 11

".˛CnC 3

2; 12/

.�C2˛CmCnC3; 12/

ˇˇ x#;

where the Pochhammer symbol notation .a/b D � .aCb/=� .a/ has been employed.

To end the exposition we obtain a derivation formula for S�;p;q.x/.

Theorem 9. For all minf<.p/;<.q/g > 0 and for <.�/ > �12

when p D q D 0,we have �

ddx

�2.S�;p;q.x//D S�;p;q.x/�

.2�C1/

2.�C1/S�C1;p;q.x/ :

Proof. Recall the series representation (3.1) of the Bessel–Struve kernel

S�;p;q.x/D� .�C1/p� �

��C 1

2

�Xn�0

B�n

2C1

2;�C

1

2Ip;q

�xn

nŠ:

With the help of the contiguous relation [7, p. 362, Theorem 3]

B.xC1;yIp;q/CB.x;yC1Ip;q/D B.x;yIp;q/ ;

a direct calculation gives�d

dx

�2.S�;p;q.x//D

� .�C1/p� �

��C 1

2

�Xn�2

B�n2�1

2;�C

1

2Ip;q

�

460 RAKESH K. PARMAR AND TIBOR K. POGANY

�B�n

2�1

2;�C

3

2Ip;q

�!xn�2

nŠ

D� .�C1/p� �

��C 1

2

�Xk�0

B�k

2C1

2;�C

1

2Ip;q

�xk

kŠ

C� .�C1/p� �

��C 1

2

�Xk�0

B�k

2C1

2;�C

3

2Ip;q

�xk

kŠ:

The rest is obvious. �

4. LOG–CONVEXITY AND TURAN TYPE INEQUALITIES

In this section, first we prove the log–convexity properties and Turan type inequal-ities for the normalized variant of modified Struve function of second kind M�;p;q.x/.Let us consider the normalized variant M�;p;qWRC 7! R, defined by

M�;p;q.x/D����C 1

2

��x2

���M�;p;q.x/ (4.1)

D2p�

Z 1

0

.1� t2/��12 e�xt�

p

t2�q

1�t2 dt: (4.2)

Theorem 10. Let minfp;qg � 0. Then the following assertions are true:� The function � 7!M�;p;q.x/ is log–convex on .�1

2;1/ for all x > 0.

� The function x 7!M�;p;q.x/ is log–convex on .0;1/ for all � > �12

.� The function .p;q/ 7!M�;p;q.x/ is log–convex on .0;1/ for all x > 0 and� > �1

2.

Moreover, for the same parameter range there holds the Turan inequality

M2�;p;q.x/�M��1;p;q.x/M�C1;p;q.x/� 0; � > �

1

2; (4.3)

which is equivalent with the Turan–type inequality

M2�;p;q.x/�M��1;p;q.x/M�C1;p;q.x/�

1

�C 12

M2�;p;q.x/; � > �

1

2: (4.4)

Furthermore, for the same parameter range there holds the Turan inequality

M2�;p;q.x/�M�;p�1;q�1.x/M�;pC1;qC1.x/� 0; � > �

1

2: (4.5)

Proof. Using the integral representation (4.2) by aid of classical Holder–Rogersinequality for integrals, we have

M��1C.1��/�2;p;q.x/D2p�

Z 1

0

.1� t2/��1C.1��/�2�12 e�xt�

p

t2�q

1�t2 dt

.p;q/–EXTENDED BESSEL–STRUVE TYPE FUNCTIONS 461

D2p�

Z 1

0

.1� t2/�.�1�12/C.1��/.�2�

12/e�xt�

p

t2�q

1�t2 dt

D2p�

Z 1

0

(.1� t2/�1�

12 e�xt�

p

t2�q

1�t2

)�

�

(.1� t2/�1�

12 e�xt�

p

t2�q

1�t2

)1��dt

�

(2p�

Z 1

0

.1� t2/�1�12 e�xt�

p

t2�q

1�t2 dt

)�

�

(2p�

Z 1

0

.1� t2/�2�12 e�xt�

p

t2�q

1�t2 dt

)1��:

This is equivalent to

M��1C.1��/�2;p;q.x/��M�1;p;q.x/

�� �M�2;p;q.x/

�1��; (4.6)

which proves the first assertion for all �1;�2 > �12 , � 2 Œ0;1� and x > 0.In a similar manner, we can prove the next two assertions that x 7!M�;p;q.x/ is

log–convex for all � > �12

, � 2 Œ0;1� and x; y > 0, that is

M�;p;q.�xC .1��/y/��M�;p;q.x/

�� �M�;p;q.y/

�1��;

while the function .p;q/ 7!M�;p;q.x/ is log–convex for all p1;p2;q1;q2 > 0, � >�12

, � 2 Œ0;1� and x > 0:

M�;�p1C.1��/p2;�q1C.1��/q2.x/�

�M�;p1;q1

.x/�� �

M�;p2;q2.x/�1��

: (4.7)

Next, choosing �1 D � � 1, �2 D �C 1 and � D 12

in (4.6) we conclude the Turaninequality (4.3).

As to the Turan–type inequality (4.4), we replace (4.1) into (4.3) and transform theresult by

� .�C 32/� .�� 1

2/

� 2.�C 12/

D2�C1

2��1; � >

1

2:

Again, specifying p1 D p� 1; q1 D q � 1, p2 D pC 1; q2 D qC 1 and � D 12

in(4.7), we deduce the Turan inequality (4.5). �

Also one proves the log–convexity of S�;p;q.x/ for x > 0.

Theorem 11. Let minfp;qg � 0. Then the following assertions are true:� The function x 7! S�;p;q.x/ is log–convex on positive real x half–axis for all� > �1

2.

462 RAKESH K. PARMAR AND TIBOR K. POGANY

� The function .p;q/ 7! S�;p;q.x/ is log–convex on .0;1/ for all x > 0 and� > �1

2.

Moreover, for the same parameter range there holds the Turan inequality

S2�;p;q.x/�S�;p�1;q�1.x/S�;pC1;qC1.x/� 0; � > �1

2; (4.8)

Proof. Considering the integral representation (3.2) assuming x;y > 0; � 2 Œ0;1�we have

S�;p;q.�xC .1��/y/�˚S�;p;q.x/

� ˚S�;p;q.y/

1��:

Indeed, this follows with the aid of the Holder–Rogers inequality for integrals, viz.Z 1

0

n.1� t2/��

12 ext�

p

t2�q

1�t2

o� n.1� t2/��

12 eyt�

p

t2�q

1�t2

o1��dt

�

(Z 1

0

.1� t2/��12 ext�

p

t2�q

1�t2 dt

)�(Z 1

0

.1� t2/��12 eyt�

p

t2�q

1�t2 dt

)1��:

completing the proof in the prescribed form.Letting p1;p2;q1;q2 > 0, � > �1

2, � 2 Œ0;1� and x > 0, we obtain

S�;�p1C.1��/p2;�q1C.1��/q2.x/�

�S�;p1;q1

.x/�� �

S�;p2;q2.x/�1��

:

Specifying here p1 D p � 1; q1 D q � 1, p2 D pC 1; q2 D qC 1 and � D 12

weconclude the Turan inequality (4.8). �

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Authors’ addresses

Rakesh K. ParmarGovernment College of Engineering and Technology, Department of Mathematics, Bikaner-334004,

Rajasthan State, IndiaE-mail address: rakeshparmar27@gmail.com

Tibor K. PoganyObuda University, Institute of Applied Mathematics, Becsi ut 96/b, 1034 Budapest, Hungary and

University of Rijeka, Faculty of Maritime Studies, Studentska 2, 51000 Rijeka, CroatiaE-mail address: poganj@pfri.hr