On Inequalities Involving Moments of Discrete Uniform Distributions

DOI: 10.15415/mjis.2012.11003

Mathematical Journal of Interdisciplinary Sciences

Vol. 1, No. 1, July 2012 pp. 35–46

©2012 by Chitkara University. All Rights

Reserved.

On Inequalities Involving Moments of Discrete Uniform Distributions

S. R. Sharma

Department of Mathematics, Chitkara University, Solan-174 103, Himachal Pradesh

R . Sharma

Department of Mathematics, H.P. University, Summer Hill, Shimla-171 005

Abstract

Some inequalities for the moments of discrete uniform distributions are obtained. The inequalities for the ratio and difference of moments are given. The special cases give the inequalities for the standard power means.

Keywords and Phrases: Moments, Power means, Discrete distribution, Geometric mean.

2000 Mathematics Subject Classification. 60 E 15, 26 D 15.

1. INTRODUCTION

Let x1, x

2, …, x

n denote n real numbers such that a ≤ x

i ≤ b, i = 1,2,…,n. The

rthorder moment µr/ of these numbers is defined as

µr ir

i

n

nx/ .=

=∑1

1

(1.1)

The power mean of order r namely Mr is defined as

Mr = µr

r/( )1

for r ≠ 0 (1.2)

and

Mr = lim /

r rr

→( )

0

1µ for r = 0. (1.3)

It may be noted that M -1, M

0 and M

1 respectively define harmonic mean,

geometric mean and Arithmetic mean. It is well known that the power mean M

r is an increasing function of r.For 0 < a ≤ x

i ≤ b, i = 1, 2,..., n, we have, [1],

µ µr s

rs/ / ,≥( ) (1.4)

where r is a positive real number and s is any real number such that r > s. If r is negative real number with r > s, the reverse inequality holds. For s = 0, the inequality (1.4) gives

Sharma, S. R. Sharma, R.

36

Mathematical Journal of Interdisciplinary Sciences, Volume 1, Number 1, July 2012

µr o

rM/ ≥( ) (1.5)

See also [2]. Some bounds for the difference and ratio of moments have also been investigated in literature; see [3-6].

Our main results give the refinements of the inequalities (1.4) and (1.5) when the minimum and maximum values of x

1 namely, a and b, and the value

of n is prescribed, (Theorem 2.1 and 2.2, below). The bounds for the difference and ratio of moments are obtained (Theorem 2.3-2.6, below). We also discuss the cases when the inequalities reduce to equalities. As the special cases, we get various bounds connecting lower order moments and the standard means of the n real numbers, (Inequalities 3.1 - 3.33, below), also see [7-9].

2. MAIN RESULTS

Theorem 2.1. For 0 < a ≤ xi ≤ b, i = 1, 2,...,n, we have

µ µr

r rr s

s

s

s srsa b

nn

na b

n/ / ,≥

++

-

-

+

-

2 (2.1)

where n ≥ 3, r is a positive real number and is any non-zero real number such that r > s For s < r < 0 the reverse inequality holds. The inequality (2.1) becomes equality when

x a x x xn a b

nx bn

ss s

n

s

1 2 3 1 2

1

= = = = =′ - -

-

=-, .

µand

The inequality (2.1) provides a refinement of the inequality (1.4).

Proof. The rth order moment of n real numbers xi, with x

1 = a and x

n = b can

be written as

µr

r r r rna b

nn

nx x

n/ ...

.=+

+-

+ +-

-22

2 1 (2.2)

Apply (1.4) to n - 2 real numbers x2, x

3,..., x

n-1, we find that

x xn

x xn

r rn

s sn

rs

2 1 2 1

2 2+ +

-≥

+ +-

- -... ..., (2.3)

where r is a positive real number and s is any non zero real number such that r > s.

Combine (2.2) and (2.3), we get

µr

r r s sn

rsa b

nn

nx x

n/ ...

.≥+

+-

+ +-

-22

2 1 (2.4)

Also,

On Inequalities Involving

Moments of Discrete Uniform

Distributions

37


′ = + + +

µs

s s s

na x b1

2 , (2.5)

Therefore

x x x n a bs sns

ss s

2 3 1+ + + = ′ - -- µ (2.6)

Substituting the value from (2.6) in (2.4), we immediately get (2.1).For s < r < 0, inequality (2.3) reverses its order [2]. It follows therefore

that inequality (2.1) will also reverse its order for s < r < 0.

Theorem 2.2. For 0 1 2< ≤ ≤ =a x b i ni , , , , , we have

µr

r ron

rna b

nn

nMab

/ ,≥+

+-

-2 2

(2.7)

where n ≥ 3 and r is a non-zero real number.The inequality (2.7) becomes equality when

x a x x x

Mab

x bnon

n

n

1 2 2 1

12

= = = = =

=-

-

, . and

The inequality (2.1) provides a refinement of the inequality (1.5).

Proof. Apply (1.5) to n - 2 real numbers x x xn2 3 1, , , , - , we find that

x x xn

x x xr r

nr

n nnn

r

2 3 12

12

3

12

1

12

2+ + +

-≥

- - ---

.... ... . (2.8)

Combine (2.2) and (2.8), we get

µr

r rn n

nn

ra b

nn

nx x x/ . .... .≥

++

-

- ---2

2

12

3

12

1

12 (2.9)

Also,

M0 = a x b n⋅ ⋅ ⋅( )2

1

... , (2.10)

therefore

x x xMabn

n

2 3 10

- = . (2.11)

Substituting the value from (2.11) in (2.9) we immediately get (2.7).

Theorem 2.3. For 0 < a ≤ xi ≤ b, i = 1,2,…,n, we have

. ./ /µ µr s

rs

r r s srs

na b a b

-( ) ≥+

-+

22 2

(2.12)


38


where r is a positive real number and s is a nonzero real number such that r > s. For s < r < 0, the reverse inequality holds.

For n = 2 inequality (2.13) becomes equality. For n ≥ 3, inequality (2.13) is sharp; equality holds when

x1 = a, x

2 = x

3 = …= x

n-1 =

a bs s1s+

2

and xn = b.

Proof. It follows from the inequality (2.1) that

µ µ µr s

rs

r rr s

s

s

s srsa b

nn

na b

n/ / /-( ) ≥

++

-

-

+

-

-

2µµs

rs

/ .( )

Consider a function

f ′( ) =+

+-

-

+

-( )

-

µ µ µs

r rr s

s

s

s srs

sa b

nn

na b

n2/ /

rrs

. (2.13)

The function f( )′µs is continuous in the interval [as, bs] and its derivative

df

d

r

s

n

n

a b

n

r

ss

r s

s

s

s sr s

s

s

r

µµ µ

// /=

-

-

+

- ( )

- -

2

--s

s . (2.14)

vanishes at

µs/ =

+a bs s

2 (2.15)

The value of the second order derivative

d f

d s/

2

2

2

2µµ=

--

-

+

- -

r

s

r s

s

n

n

a b

n

r s

s

s

s sr

/

ss

s

s

r s

s-( )

-

µ / ,2

(2.16)

at ′ =+

µs

s sa b

2 is

d f

d

r r s

s n

a b

s

s sr s

s2

2 2

2

2

2 2µ /

( )=

--

+

-

(2.17)

It follows from (2.17) that

d f

d s/

2

2µ > 0 for r > 0



Distributions

39


and

d f

d s/

2

2µ < 0 for r < 0.

Therefore, for r > 0, the function f ′( )µs achieves its minimum when the value of ′µs is given by (2.15). Hence

. fn

a b a bs

r r s sr

s

′( )≥+

-

+

µ2

2 2.

This proves (2.12).Likewise, for r < 0, the function f ′( )µs achieves its maximum when the

value of ′µs is given by (2.15) and we conclude that

µ µr s

r

s

r r s sr

s

n

a b a b/ / .-( ) ≤+

-+

2

2 2

Theorem 2.4. For 0 < a ≤ xi ≤ b, i = 1, 2,…, n, we have

µrr

r r r

Mn

a bab/ - ≥

+-( )

0

2

2 (2.18)

where r is a non zero real number. For n = 2, the inequality (2.19) becomes equality. For n ≥ 3, inequality

(2.19) is sharp, equality holds when

x1 = a, x

2 = x

3 = … = x

n-1 = ab and x

n = b .

Proof. It follows from the inequality (2.7) that

µrr

r r nr

nrM

a b

n

n

n

M

abM/ - ≥

++

-

--

00

2

0

2.

Consider a function

g(M0) = a b

n

n

n

M

abM

r r nr

nr+

+-

--2 0

2

0. (2.19)

The derivative

dg

dMrM

M

abr

r

n

00

1 0

2

2

1=

-

-- (2.20)


40


vanishes at

M ab0 = (2.21)

The value of the second derivative

d g

dMrM

n r

n

M

abrr

r

n2

00

2 0

2

21 2

21=

- +-

- -

--( )

( ). (2.22)

at M ab0 = is

d g

dM

r

nab

r2

02

2 2

22

2=

-

-

( ) .

Clearly,

d g

dM

2

02

> 0 for M ab0 = .

Therefore, for r ≠ 0, the function g(M0) attains its minimum at M ab0 = and

we have

gn

a babs

r r r′( )≥

+-( )

µ

2

2.

This proves (2.18).

Theorem 2.5. For 0 < a ≤ xi ≤ b for i = 1,2,…,n, we have

µ

µ

r

s

rs

r r

s ss s

r r

s s

sra b

a b na b n a b

a b

/

/( )

( )≥

++

+ + -++

1 2--

-

s

s rs

. (2.23)

where r is a positive real number and s is any non-zero real number such that r > s.

For s < r < 0, the reverse inequality holds. For n = 2, the inequality (2.23) becomes equality. For n ≥ 3, this inequality

is sharp; equality

holds when x1 = a, x

2 = x

3 = …= x

n-1=

a b

a b

r r

s s

r-s++

1

and xn = b.



Distributions

41


Proof. It follows from inequality (2.1) that

µ

µµr

s

rs

r rr s

s

s

s sa b

n

n

n

a b

n

/

/

/

( )≥

++

-

-

+

-

2

rrs

s

rs

( )

1

µ /, (2.24)

where r is a positive real number and s is a non zero real number such that r > s. We now find the minimum value of the right side expression in (2.24) as ′µs varies over the interval [as,bs].

Consider a function

ha b

n

n

n

a b

ns

r rr s

s

s

s sr

s

′( )=+

+-

′ -

+

-

µ µ2

′( )

1

µs

r

s

. (2.25)

The function h( ′µs ) is continuous in the interval [as, bs] and its derivative

dh

d

r a b

sn

n a b

ns

s s

s

r s

ss

s s

µ µµ

/ /

/( )=

+

- --

+

1

2

-++

-r s

s r r

s s

a b

a b (2.26)

vanishes at

′ =+

+- +

+

-

µs

s s r r

s s

sr sa b

nn

na ba b

2. (2.27)

From (2.26), if r is a positive real number and s is any non zero real

numbersuch that r > s then the sign of dh

d s′µ changes from negative to positive

while ′µs passes through the value given by (2.27). The function h( ′µs )

achieves its minimum at the value of ′µs given by (2.27). We therefore have,

h a ba b n

a b n a ba bs

r r

s ss s

r r

s s

sr s

′( ) ≥++

+ + -( ) ++

-

µ1 2

-s rs

. (2.28)

Combining (2.24), (2.25) and (2.28), the inequality (2.24) follows immediately.

For s < r < 0, it follows from (2.26) that the sign of dhd s′µ

changes from

positive to negative while ′µs passes through the value given by (2.27). In this

case function f( ′µs ) attains its maximum at the value of ′µs given by (2.27). We therefore have,


42


h a ba b n

a b n a ba bs

r r

s ss s

r r

s s

sr s

′( ) ≤++

+ + -( ) ++

-

µ1 2

-s rs

. (2.29)

On the other hand we conclude from theorem 2.1 that

µ

µµr

s

rs

r rr s

s

s

s sa bn

nn

a bn

/

/

/

( )≤

++

-

-

+

-

2

rrs

s

rs

( )

1

µ /. (2.30)

Combining (2.25), (2.26) and (2.27), we find that for s < r < 0,

′

′( )≤

++

-

′ -

+

-

µ

µµr

s

rs

r rr s

s

s

s sa bn

nn

a bn2

rrs

s

rs

′( )

1

µ.

Theorem 2.6 For 0 < a ≤ xi ≤ b for i = 1,2,…,n, we have

µrr

r r

r

n

Ma b

ab

/

0

2

2≥

+

( )

(2.31)

where r is a non zero real number.For n = 2, the inequality (2.31) becomes equality. For n ≥ 3, the inequality

(2.31) is sharp; equality holds when

x1 = a, x

2 = x

3= …= x

n-1=

a br r1r+

2

and xn = b.

Proof. It follows from inequality (2.7) that

F M a b nn

Mab M

r r nr

n

r00

2

022 1( ) =

++

-

-

. (2.32)

where r is a non zero real number. We now find the minimum value of the right hand side expression in (2.33).

Consider a function

µrr

r ron

rn

orM

a bn

nn

Mab M

/

0

22 1≥

++

-

- (2.33)

The derivative

dFdM

rnM

Mab

a b

or

on

rn r r

01

222

=

-+

+

-

. (2.34)



Distributions

43


vanishes at

M ab a bnr r

nnr

0

12

2=

+

-

( ) . (2.35)

From (2.34), if r is a non zero real number then the sign of dFdM 0

changes

fromnegative to positive while Mo passes through the value given by (2.35).

The function f(Mo) attains its minimum at the value of M

o given by (2.35). We

therefore have

F M a b

ab

r r

r

n

0

2

2( )≥

+

( )

. (2.36)

Combining (2.32), (2.33) and (2.36), inequality (2.31) follows immediately.

3. SOME SPECIAL CASES

From the application point of view, it is of interest to know the bounds for the first four moments and the bounds for standard power means (namely, Arithmetic mean, (µ1

/ = x ), Geometric mean (G) and Harmonic mean (H)). These bounds are also of fundamental interest in the theory of inequalities. By assigning particular values to r and s, in the generalized inequalities obtained in this paper, we can find inequalities connecting various power means and moments. The following inequalities can be deduced easily from the generalized inequalities given in Theorems 2.1 and 2.2:

µ µ2

2 2

1

2

2/ /≥

++

--

+

a bn

nn

a bn

, (3.1)

µ2

2 2 3 22/

( )≥

++

-

- +

a bn

nn

abnHabn a b H

, (3.2)

µ2

2 22

22/ ≥+

+-

-a bn

nn

Gab

n n

, (3.3)

µ1

22/

( )≥

++

-

- +

a bn

nn

abnHabn a b H

, (3.4)

µ1

122/ ≥

++

-

-a bn

nn

Gab

n n, (3.5)


44


G ≥-

-+

-

( )abn

n

H

a b

ab

n

n

n

1

2

2, (3.6)

µ µ3

3 3 2

1

3

2/ /≥

++

-

-

+

a bn

nn

a bn

, (3.7)

µ µ3

3 3

2

2 232

2/ /≥

++

--

+

a bn

nn

a bn

, (3.8)

µ µ4

4 4 3

1

4

2/ /≥

++

-

-

+

a bn

nn

a bn

, (3.9)

µ µ4

4 4

2

2 2 2

2/ /≥

++

--

+

a bn

nn

a bn

(3.10)

and µ µ4

4 43

3

3 3

2

43

/ /≥+

+-

-+

a bn

nn

a bn

. (3.11)

The following lower bounds for the difference of moments and means are deduced from the generalized inequalities given in Theorems 2.3 and 2.4:

µ µ2 12 21

2/ /- ≥ -( )

nb a , (3.12)

µ22

2

21 2/ ( )( )

- ≥-

++

H b a

nab

a b, (3.13)

µ22

2/ ( )- ≥

-G b an

, (3.14)

µ1

2/

( )- ≥

-( )+

Hb a

n a b, (3.15)

µ1

2/ ( )- ≥

-G b an

, (3.16)

G H- ≥

+

( ) -

+

-

2 22

22

ab

a bab

ab

a b

n

n n n

, (3.17)

µ µ3 13 23

4/ /- ≥

+-( )a b

nb a , (3.18)



Distributions

45


µ µ3 2

32

3 3 2 23

222 2

/ /- ≥+

-+

n

a b a b, (3.19)

µ µ4 14

4 4 422 2

/ /- ≥+

-+

n

a b a b, (3.20)

µ µ4 22

4 4 2 2 22

2 2/ /- ≥

+-

+

n

a b a b (3.21)

and

µ µ4 3

43

4 4 3 34

322 2

/ /- ≥+

-+

n

a b a b. (3.22)

The following lower bounds for the ratio of moments and means are deduced from the generalized inequalities given in Theorems 2.5 and 2.6:

µµ

2

12

2 2

2 2 22

/

/

( )( ) ( )( )

≥+

+ + - +a b n

a b n a b, (3.23)

µµ

3

13

2 2 2

2 22

2

/

/

( )

( )≥

- +

+ + - - +

a ab b n

a b n a ab b, (3.24)

µ

µ3

2

32

3 3

2 2 3 3 3 21

22

/

/

( )

( ) ( )( )≥

+

+ + - +

n a b

a b n a b, (3.25)

µµ

4

14

3 4 4

43 4 4 1

33

2

/

/

( )

( ) ( )( )≥

+

+ + - +

n a b

a b n a b

, (3.26)

µµ

4

22

4 4

2 2 2 4 42

/

/

( )( ) ( )( )

≥+

+ + - +n a b

a b n a b, (3.27)

µ

µ4

3

43

3 4 4

3 3 4 4 4 31

32

/

/

( )

( ) ( )( )≥

+

+ + - +

n a b

a b n a b, (3.28)

µ12

21 2

/

H nn a b

ab≥ - +

+

, (3.29)


46


µ1

2

2

/

Ga b

ab

n≥

+

, (3.30)

GH

a bab

n≥

+

2

2

, (3.31)

µ22

2 22

2

/

Ga b

ab

n≥

+

(3.32)

and

µ22 3

2 2 2

2 2

13

3

1 2/ ( )( )

H na b a b

a bn≥

+ +

+ -

. (3.33)

ACKNOwLEDgEMENT

The second author acknowledges the support of the UGC-SAP.

REFERENCES

[1] G.H. Hardy, J.E. Littlewood and G. Polya, Inequalities, New York, Cambridge University Press, Cambridge, U.K. (1934).

[2] R. Sharma, R,G, Shandil, S. Devi and M. Dutta, Some Inequalities between Moments of probability distribution, J. Inequal. Pure and Appl. Math., 5 (2): Art. 86, (2004).

[3] B.C. Rennie, On a Class of Inequalities, J. Austral. Math. Soc., 3: 442–448, (1963).[4] S. Ram, S. Devi, and R. Sharma, Some inequalities for the ratio and difference of moments,

International Journal of Theoretical and Applied Sciences, 1(1): 103–110, (2009). http://dx.doi.org/10.1017/S1446788700039057[5] R. Sharma, A. Kaura, M. Gupta and S. Ram, Some bounds on sample parameters with refinements

of Samuelson and Brunk inequalities. Journal of Mathematical Inequalities, 3:99–106, (2009). http://dx.doi.org/10.7153/jmi-03-09[6] R. Sharma, Some more inequalities for arithmetic mean, harmonic mean and variance, Journal

of Mathematical Inequalities, 2: 109–114, (2008). http://dx.doi.org/10.7153/jmi-02-11[7] E.N. Laguerre, Surune mathode pour obtenir par approximation les raciness d’ une Equation

algebrique qul a tautes ses raciness [in French], Nouvelles Annales de Mathematiques (Paris), 2e Serie 19: 161–171 and 193–202, (1980) [JFM12:71].

[8] A. Lupas, Problem 246, Mathematicki vesnik (Belgrade), 8 (23): 333, (1971).[9] S.T. Jensen and G.P.H. Styan, Some comments and a bibliography on the Laguerre Samuelson

inequalities with extensions and applications to statistics and matrix theory, Analytic and Geometric inequalities and their applications (Hari M. Srivastava and Themistocles M. Rassias, eds), Springer-Verlag, (1999).

SitaRam Sharma is Assistant Professor Department of Applied Sciences, Chitkara University Himachal Pradesh.

Rajesh Sharma is Associate Professor, Department of Mathematics Himachal Pradesh.

On Inequalities Involving Moments of Discrete Uniform Distributions

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