Positive Semidefinite Matrices, Exponential Convexity for Majorization, and Related Cauchy Means

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Volume 2010, Article ID 728251,19pages doi:10.1155/2010/728251

Research Article

Positive Semidefinite Matrices,

Exponential Convexity for Majorization, and

Related Cauchy Means

M. Anwar,

1

_{N. Latif,}

2

_{and J. Pe ˇcari ´c}

2, 3

1_{Center for Advanced Mathematics and Physics, NUST Campus at College of}

Electrical & Mechanical Engineering, Peshawar Road, Rawalpindi 48000, Pakistan 2_{Abdus Salam School of Mathematical Sciences, GC University, Lahore 5400, Pakistan}

3_{Faculty of Textile Technology, University of Zagreb, 1002 Zagreb, Croatia}

Correspondence should be addressed to M. Anwar,matloob [email protected]

Received 26 October 2009; Accepted 10 March 2010

Academic Editor: Panayiotis Siafarikas

Copyrightq2010 M. Anwar et al. This 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 prove positive semidefiniteness of matrices generated by differences deduced from majorization-type results which implies exponential convexity and log-convexity of these differences and also obtain Lyapunov’s and Dresher’s inequalities for these differences. We introduce new Cauchy means and show that these means are monotone.

1. Introduction and Preliminaries

Let x x1, . . . , xn, p p1, . . . , pn denote two sequences of positive real numbers with

_n

i1pi 1. The well-known Jensen inequality for convex function1, page 43gives that, for

t <0 ort >1,

n

i1

pix_it≥

_n

i1

pixi

t

, 1.1

and vice versa for 0< t <1.

In2, the following generalization of this theorem is given.

Theorem 1.1. For−∞< r < s < t <∞,

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where

λt:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

_n

i1pixit−

_n

i1pi xi

t

tt−1 , t /0,1,

log

_n

i1

pixi

−n

i1

pilogxi, t0,

n

i1

pixilogxi−

_n

i1

pixi

log

_n

i1

pixi

, t1.

1.3

For fixedn≥2 let

x x1, . . . , xn, y

y1, . . . , yn

1.4

denote twon-tuples. Let

x1≥x2≥ · · · ≥xn, y1≥y2≥ · · · ≥yn,

x1≤x2≤ · · · ≤xn, y1≤y2≤ · · · ≤yn

1.5

be their ordered components.

Definition 1.2see1, page 319. yis said to majorizexorxis said to be majorized byy, in

symbol,yx, if

m

i1

xi≤ m

i1

yi 1.6

holds form1,2, . . . , n−1 and

n

i1

xi n

i1

yi. 1.7

Note that1.6is equivalent to

n

in−m1

xi≤ n

in−m1

yi 1.8

form1,2, . . . , n−1.

The following theorem is well-known as the majorization theorem and a convenient

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Theorem 1.3. Let I be an interval inR, and let x, y be twon-tuples such thatxi,yi ∈ I i

1, . . . , n. Then

n

i1

φxi≤n

i1

φyi

1.9

holds for every continuous convex functionφ:I → Rif and only ifyxholds.

Remark 1.4see4. If φxis a strictly convex function, then equality in1.9is valid iﬀ

xiyi, i1, . . . , n.

The following theorem can be regarded as a generalization of Theorem 1.3 and is

proved by Fuchs in5 see also1, page 323.

Theorem 1.5. Letx, ybe two decreasing real n-tuples, and letp p1, . . . , pnbe a realn-tuple such

that

k

i1

pixi≤ k

i1

piyi fork1, . . . , n−1,

n

i1

pixi n

i1

piyi.

1.10

Then for every continuous convex functionφ:I → R, one has

n

i1

piφxi≤ n

i1

piφ

yi

. 1.11

Definition 1.6. A functionh:a, b → Ris exponentially convex function if it is continuous and

n

i,j1

ξiξjhxixj≥0 1.12

for all n ∈ Nand all choices ξi ∈ Rand xi ∈ a, b,i 1, . . . , nsuch thatxixj ∈ a, b,

1≤i, j≤n.

The following proposition is given in6.

Proposition 1.7. Leth:a, b → R. The following propositions are equivalent.

ihis exponentially convex.

iihis continuous and

n

i,j1

ξiξjh

_x

ixj

2

≥0, 1.13

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Corollary 1.8. Ifhis exponentially convex, then

det

h

_x

ixj

2

n

i,j1

≥0, 1.14

for everyn∈Nand everyxi∈a, b,i1, . . . , n.

Corollary 1.9. Ifh:a, b → Ris exponentially convex function, thenhis alog-convex function in Jensens sense:

h

_x_y

2

≤hxhy, ∀x, y∈a, b. 1.15

In this paper, we prove positive semidefiniteness of matrices generated by diﬀerences

deduced from majorization-type results 1.9, 1.11, 4.2, and 4.5 which implies

exponential convexity and log-convexity of these diﬀerences and also obtain Lyapunov’s and

Dresher’s inequalities for these diﬀerences. In 7, new Cauchy means are introduced. By

using these means, a generalization of1.2was givensee7. In the present paper, we give

related results in discrete and indiscrete cases and some new means of the Cauchy type.

2. Main Results

Lemma 2.1. Define the function

ϕsx:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

xs

ss−1, s /0,1, −logx, s0,

xlogx, s1.

2.1

Thenϕ _sxs−2_{, that is,}_ϕ

sis convex forx >0.

Definition 2.2. It is said that a positive functionfis log-convex in the Jensen sense on some

intervalI⊆Rif

fsft≥f2

st

2

2.2

holds for everys, t∈I.

The following lemma gives an equivalent condition for convexity of function f 1,

page 2.

Lemma 2.3. Ifφis convex on an intervalI ⊆R, then

φs1s3−s2 φs2s1−s3 φs3s2−s1≥0 2.3

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Theorem 2.4. Letxandybe two positiven-tuples,yx,

Λt Λtx;y :

n

i1

ϕt

yi

−n

i1

ϕtxi, 2.4

and allxi’s andyi’s are not equal.

Then the following statements are valid.

aFor everyn ∈ Nands1, . . . , sn ∈R, the matrixΛsisj/2

n

i,j1 is a positive semidefinite

matrix. Particularly

detΛsisj/2

k

i,j1 ≥0 2.5

fork1, . . . , n.

bThe functions→Λsis exponentially convex.

cThe functions→Λsislog-convex onRand the following inequality holds for−∞< r < s < t <∞:

Λt−r

s ≤Λtr−s Λst−r. 2.6

Proof. aConsider the function

μx k i,j

uiujϕsijx 2.7

fork1, . . . , n,x >0,ui∈R,sij ∈R, wheresij sisj/2 andϕsijis defined in2.1. We have

μ x k

i,j

uiujxsij−2

_k

i

uixsi/2−1

2

≥0, x≥0. 2.8

This shows thatμis a convex function forx≥0.

UsingTheorem 1.3,

n

m1

μym

−n

m1

μxm≥0. 2.9

This implies that

n

m1

⎛

⎝k

i,j

uiujϕsij

ym

⎞

⎠₋n

m1

⎛

⎝k

i,j

uiujϕsijxm

⎞

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or equivalently

k

i,j

uiujΛsij ≥0. 2.11

From last inequality, it follows that the matrixΛsisj/2

n

i,j1is a positive semidefinite matrix,

that is,2.5is valid.

b Note that Λs is continuous for s ∈ R. Then by using Proposition 1.7, we get

exponentially convexity of the functions → Λs.

cSinceϕtxis continuous and strictly convex function forx >0 and allxi’s and

yi’s are not equal, therefore byTheorem 1.3withφϕtwe have

n

i1

ϕt

yi

> n

i1

ϕtxi. 2.12

This implies

Λt Λtx;y

n

i1

ϕtyi− n

i1

ϕtxi>0, 2.13

that is,Λtis a positive-valued function.

A simple consequence ofCorollary 1.9is thatΛsis log-convex; then by definition

logΛt_s−r ≤logΛ_rt−slogΛs_t−r, 2.14

which is equivalent to2.6.

Theorem 2.5. LetΛtbe defined as inTheorem 2.4andt, s, u, v∈Rsuch thats≤u,t≤v,_{s /}t, and

u /v. Then

_Λt

Λs

1/t−s

≤

_Λv

Λu

1/v−u

. 2.15

Proof. For a convex functionϕ, a simple consequence of2.3is the following inequality1,

page 2:

ϕx2−ϕx1

x2−x1 ≤

ϕy2

−ϕy1

y2−y1

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withx1≤y1, x2 ≤y2, x1/x2, y1/y2. Since byTheorem 2.4candΛtis log-convex, we can

set in2.16ϕx logΛt, x1s, x2 t, y1u, andy2v. We get

logΛt−logΛs

t−s ≤

logΛv−logΛu

v−u , 2.17

from which2.15follows.

Theorem 2.6. Letxandybe two positive decreasingn-tuples, letp p1, . . . , pnbe a realn-tuple

and let

λtλtx,y;p : n

i1

piϕt

yi

−n

i1

piϕtxi 2.18

such that conditions1.10are satisfied andλtis positive. Then the following statements are valid.

aFor everyn ∈ Nand s1, . . . , sn ∈ R, the matrixλsisj/2

n

detλsisj/2

k

i,j1≥0 2.19

fork1, . . . , n.

bThe functions→λsis exponentially convex.

cThe functions→λsislog-convex onRand the following inequality holds for−∞ < r < s < t <∞:

λt_s−r ≤λ_rt−sλs_t−r. 2.20

Proof. As in the proof ofTheorem 2.4, we useTheorem 1.5instead ofTheorem 1.3.

Theorem 2.7. Letλtbe defined as inTheorem 2.6andt, s, u, v∈Rsuch thats≤u,t≤v,_{s /}t, and

u /v. Then

λt λs

1/t−s

≤

λv λu

1/v−u

. 2.21

Proof. Similar to the proof ofTheorem 2.5.

3. Cauchy Means

Let us note that2.15and2.21have the form of some known inequalities between means

e.g., Stolarsky means, Gini means, etc. Here we will prove that expressions on both sides of

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Lemma 3.1. Letf ∈C2_I_{, I interval in}_R_{, be such that}

m≤f x≤M. 3.1

Consider the functionsφ1,φ2defined as

φ1x Mx 2

2 −fx,

φ2x fx−

mx2

2 ,

3.2

thenφixfori1,2are convex.

Proof. Since

φ ₁x M−f x≥0,

φ ₂x f x−m≥0, 3.3

that is,φifori1,2 are convex.

Denote

I1 m1, M1, wherem1min

mx, my, M1max

Mx, My. 3.4

In the above expression,m_xandm_yare the minimums ofxandy, respectively. Similarly,M_x

andMyare the maximums ofxandyrespectively.

Theorem 3.2. Letx and ybe two positive n-tuples,y x, allxi’s andyi’s are not equal, and f∈C2_I

1,withI1being defined as in3.4, then there existsξ∈I1such that

n

i1

fyi

−n

i1

fxi f ξ

2

_n

i1

yi2− n

i1

x2_i

. 3.5

Proof. Sincef ∈C2_I

1andI1is compact, thenm1≤f x≤M1forx∈I1.Then by applying

φ1andφ2defined inLemma 3.1forφinTheorem 1.3, we have

n

i1

φ1xi≤

n

i1

φ1

yi,

n

i1

φ2xi≤

n

i1

φ2

yi,

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that is,

n

i1

fyi− n

i1

fxi≤ M1

2

_n

i1

yi2− n

i1

x2_i

, 3.7

n

i1

fyi

−n

i1

fxi≥ m1

2

_n

i1

yi2− n

i1

x2_i

. 3.8

By combining3.7and3.8

m1≤2

_n

i1f

yi

−n i1fxi

n

i1yi2−

n

i1xi2

≤M1. 3.9

_n

i1yi2−

_n

i1xi2/0 because xi/yi, fori 1, . . . , nby usingRemark 1.4. Using the fact

that form1≤ρ≤M1, there existsξ∈I1such thatf ξ ρ, we get3.5.

Theorem 3.3. Let x and y be two positive n-tuples, y x, all xi’s and yi’s are not equa,and f, g∈C2_I

1, withI1being defined as in3.4, then there existsξ∈I1such that

_n

i1f

yi

−n i1fxi

_n

i1g

yi

−n i1gxi

f ξ

g ξ, 3.10

provided thatg x/0for everyx∈I1.

Proof. Let a functionk∈C2_I

1be defined as

kc1f−c2g, 3.11

wherec1andc2are defined as

c1

n

i1

gyi

−n

i1

gxi,

c2

n

i1

fyi− n

i1

fxi.

3.12

Then, usingTheorem 3.2withfk, we have

0

c1

f ξ

2 −c2

g ξ

2

n

i1

yi2− n

i1

xi2

. 3.13

Since

n

i1

yi2− n

i1

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therefore,3.13gives

c2

c1

f ξ

g ξ. 3.15

After putting values, we get3.10. The denominator of left-hand side is nonzero by using

fginTheorem 3.2.

Corollary 3.4. Letxandybe two positiven-tuples such thaty xand allxi’s andyi’s are not

equal, then for−∞< s /t /0, 1/s <∞there existsξ∈I1, withI1being defined as in3.4, such

that

ξt−s ss−1 tt−1

n

i1yit−

n

i1xit

n

i1yis−

n

i1xis

. 3.16

Proof. Settingfx xt_and_gx_xs_,_{t /}_{s /}₀_,_{1 in}_3.10_{, we get}_3.16_.

Remark 3.5. Since the functionξ→ξt−s_{is invertible, then from}_3.16_{we have}

m1 ≤

ss−1

tt−1

n

i1yit−

n

i1xit

n

i1yis−

n

i1xis

1/t−s

≤M1. 3.17

In fact, similar result can also be given for 3.10. Namely, suppose thatf /g has

inverse function. Then from3.10, we have

ξ

_f

g

−1n

i1f

yi−n_i₁fxi

n

i1g

yi−n_i₁gxi

. 3.18

So, we have that the expression on the right-hand side of 3.18 is also a mean. By the

inequality3.17, we can consider for positiven-tuplesxandysuch thatyx,

Mt,s

_Λt

Λs

1/t−s

3.19

for−∞ < s /t < ∞, as means in broader sense. Moreover we can extend these means in

other cases. So passing to the limit, we have

Ms,sexp

n

i1yislogyi−

_n

i1xislogxi

_n

i1yis−

_n

i1xis

− 2s−1

ss−1

, s /0,1,

M0,0exp

n

i1log2yi−

_n

i1log2xi

2n_i₁logyi−

_n

i1logxi

1

,

M1,1exp

n

i1yilog2yi−

_n

i1xilog2xi

2n_i₁yilogyi−

_n

i1xilogxi

−1

.

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Theorem 3.6. Lett, s, u, v∈Rsuch thatt≤u,s≤v, then the following inequality is valid:

Mt,s≤Mu,v. 3.21

Proof. SinceΛsis log-convex, therefore by2.15we get3.21.

Theorem 3.7. Let x and y be two positive decreasingn-tuples, let p be a real n-tuple such that conditions1.10are satisfied,λtis positive defined as inTheorem 2.6, andf∈C2_I

1,withI1being

defined as in3.4, then there existsξ∈I1such that

n

i1

pifyi− n

i1

pifxi f ξ

2

_n

i1

piyi2− n

i1

pix2_i

, 3.22

Theorem 3.8. Letxandybe two positive decreasingn-tuples,pbe a realn-tuple such that conditions

1.10are satisfied,λtis positive defined as inTheorem 2.6andf, g∈C2_I

1,I1is defined as in3.4.

Then there existsξ∈I1such that

_n

i1pif

yi

−n

i1pifxi

_n

i1pig

yi

−n

i1pigxi

f ξ

g ξ. 3.23

provided thatg x/0for everyx∈I1.

Corollary 3.9. Let x and y be two positive decreasing n-tuples, let p be a real n-tuple such that conditions 1.10 are satisfied and λt is positive defined as in Theorem 2.6, then for −∞ < s /t /0,1/s <∞there existsξ∈I1, withI1being defined as in3.4, such that

_n

i1piyit−

_n

i1pixit

_n

i1piyis−

_n

i1pixis

. 3.24

Proof. Settingfx xt_and_gx_xs_, _{t /}_{s /}₀_,_{1 in}_3.23_{, we get}_3.24_.

m1 ≤

ss−1

tt−1

n

i1piyit−

n

i1pixit

_n

i1piyis−

_n

i1pixis

1/t−s

≤M1. 3.25

ξ

_f

g

−1n

i1pif

yi

−n

i1pifxi

_n

i1pig

yi

−n

i1pigxi

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inequality3.25, we can consider for positive n-tuplesx andysuch that conditions 1.10

are satisfied, and

Mt,s

λt λs

1/t−s

3.27

Ms,sexp

n

i1pi yislogyi−

_n

i1pixislogxi

_n

i1piyis−

_n

i1pixis

− 2s−1

ss−1

, s /0,1.

M0,0exp

n

i1pilog2yi−

_n

i1pilog2xi

2 n_i₁pilogyi−

_n

i1pilogxi

1

.

M1,1exp

n

i1piyilog2yi−

_n

i1pixilog2xi

2n_i₁piyilogyi−

_n

i1pixilogxi

−1

.

3.28

Mt,s≤Mu,v. 3.29

Proof. Sinceλsis log-convex, therefore by2.21we get3.29.

4. Some Related Results

Let xτ, yτ be real valued functions defined on an intervala, b such that _asxτdτ,

_s

ayτdτboth exist for alls∈a, b.

Definition 4.1see1, page 324. yτis said to majorizexτ, in symbol,yτ xτ, for

τ ∈a, bif they are decreasing inτ∈a, band

_s

a

xτdτ≤

_s

a

yτdτ fors∈a, b, 4.1

and equality in4.1holds forsb.

The following theorem can be regarded as a majorization theorem in integral case1,

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Theorem 4.2. yτxτforτ∈a, biﬀthey are decreasing ina, band

_b

a

φxτdτ ≤

_b

a

φyτdτ 4.2

holds for everyφthat is continuous, and convex ina, bsuch that the integrals exist.

The following theorem is a simple consequence of Theorem 12.14 in8 see also1,

page 328:

Theorem 4.3. Letxτ, yτ :a, b → R,xτandyτare continuous and increasing and let

G:a, b → Rbe a function of bounded variation.

aIf

_b

ν

xτdGτ≤

_b

ν

yτdGτ for everyν∈a, b, 4.3

_b

a

xτdGτ

_b

a

yτdGτ 4.4

hold, then for every continuous convex functionf, one has

b

a

fxτdGτ≤

b

a

fyτdGτ. 4.5

bIf 4.3holds, then4.5holds for every continuous increasing convex functionf.

Theorem 4.4. Letxτandyτbe two positive functions defined on an intervala, b, decreasing ina, b,yτxτ,

βtxτ;yτ:

b

a

ϕtyτdτ−

b

a

ϕtxτdτ, 4.6

andβtis positive.

Then the following statements are valid.

aFor everyn ∈ Nand s1, . . . , sn ∈ R, the matrixβsisj/2

n

detβsisj/2

k

i,j1≥0 4.7

fork1, . . . , n.

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cThe functions→βsislog-convex onRand the following inequality holds for−∞< r < s < t <∞:

βt_s−r ≤β_rt−sβs_t−r. 4.8

Theorem 4.5. Letβtbe defined as inTheorem 4.4andt, s, u, v∈Rsuch thats≤u,t≤v,s /t, and

u /v. Then

_βt

βs

1/t−s

≤

_βv

βu

1/v−u

. 4.9

Denote

I2 m2, M2, wherem2min

mxτ, myτ, M2max

Mxτ, Myτ. 4.10

In the above expression,mxτ andmyτ are the minimums ofxτand yτ, respectively.

Similarly,MxτandMyτare the maximums ofxτandyτ, respectively.

Theorem 4.6. Letxτ andyτbe two positive decreasing functions ina, bsuch thatyτ

xτ,βtis positive defined as inTheorem 4.4, andf∈C2I2, withI2being defined as in4.10, then

there existsξ∈I2such that

b

a

fyτdτ−

b

a

fxτdτ f ξ

2

b

a

y2τdτ−

b

a

x2τdτ

. 4.11

Theorem 4.7. Letxτ andyτbe two positive decreasing functions ina, bsuch thatyτ

xτ,βtis positive defined as inTheorem 4.4, andf, g∈ C2_I

2, withI2being defined as in4.10.

Then there existsξ∈I2such that

b

af

yτdτ−_abfxτdτ

b

ag

yτdτ−_abgxτdτ f ξ

g ξ, 4.12

provided thatg z/0for everyz∈I2.

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Corollary 4.8. Letxτand yτbe two positive decreasing functions ina, bsuch thatyτ

xτand βtis positive defined as inTheorem 4.4, then for−∞ < s /t /0,1/s < ∞there exists ξ∈I2, withI2being defined as in4.10, such that

b

aytτdτ−

b

axtτdτ

b

aysτdτ−

b

axsτdτ

. 4.13

Proof. Setfx xt_and_gx_xs_,_{t /}_{s /}₀_,_{1 in}_4.12_{, we get}_4.13_.

m2≤

⎧ ⎨ ⎩

ss−1

tt−1

_b

aytτdτ−

_b

axtτdτ

_b

aysτdτ−

_b

axsτdτ

⎫ ⎬ ⎭

1/t−s

≤M2. 4.14

ξ

_f

g

−1⎛

⎝

_b

af

yτdτ−_abfxτdτ

b

ag

yτdτ−_abgxτdτ

⎞

⎠_. _4.15

inequality4.14, we can consider for positive functionsxτandyτsuch thatyτxτ,

and # Mt,s _β t βs

1/t−s

4.16

#

Ms,sexp

⎛ ⎝

_b

aysτlogyτdτ−

_b

axsτlogxτdτ

_b

aysτdτ−

_b

axsτdτ

−_ss2s−₋1 1

⎞

⎠_, _{s /}₀_,₁_,

#

M0,0 exp

⎛ ⎜ ⎝

_b

alog

2_yτdτ₋b alog

2_xτdτ

2 _ablogyτdτ−_ablogxτdτ

1

⎞ ⎟

⎠,

#

M1,1 exp

⎛ ⎜ ⎝

b

ayτlog

2_yτ_dτ₋b

axτlog

2_xτ_dτ

2 _abyτlogyτdτ−_abxτlogxτdτ −

1

⎞ ⎟

⎠.

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#

Mt,s≤M#u,v. 4.18

Proof. Sinceβsis log-convex, therefore by4.9we get4.18.

Theorem 4.11. Letxτ, yτ:a, b → R,xτandyτare positive, continuous, and increasing and letG:a, b → Rbe a function of bounded variation. Also let

Γtxτ, yτ; Gτ:

_b

a ϕt

yτdGτ−

_b

a

ϕtxτdGτ, 4.19

such that conditions4.3and4.4are satisfied andΓtis positive. Then the following statements are valid.

aFor everyn ∈ Nand s1, . . . , sn ∈ R, the matrixΓsisj/2

n

detΓsisj/2

k

i,j1≥0 4.20

fork1, . . . , n.

bThe functions→Γsis exponentially convex.

cThe functions→Γsislog-convex onRand the following inequality holds for−∞< r < s < t <∞:

Γt−r

s ≤Γtr−sΓst−r. 4.21

Theorem 4.12. LetΓtbe defined as inTheorem 4.11andt, s, u, v∈Rsuch thats ≤u,t≤v,s /t, andu /v. Then

_Γt

Γs

1/t−s

≤

_Γv

Γu

1/v−u

. 4.22

Theorem 4.13. Letxτand yτbe positive, continuous, and increasing functions ina, bsuch that conditions4.3and4.4are satisfied,Γtis positive defined as inTheorem 4.11,f ∈C2_I

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I2being defined as in4.10, andG:a, b → Rbe a function of bounded variation, then there exists

ξ∈I2such that

b

a

fyτdGτ−

b

a

fxτdGτ f ξ

2

b

a

y2τdGτ−

b

a

x2τdGτ

. 4.23

Theorem 4.14. Letxτand yτbe positive, continuous and increasing functions in a, bsuch that conditions4.3and4.4are satisfied,G:a, b → Rbe a function of bounded variation,Γtis positive defined as inTheorem 4.11, andf, g∈C2_I

2, withI2being defined as in4.10. Then there

existsξ∈I2such that

_b

af

yτdGτ−b

afxτdGτ

b

ag

yτdGτ−b

agxτdGτ

f ξ

g ξ, 4.24

provided thatg z/0for everyz∈I2.

Corollary 4.15. Letxτandyτbe positive, continuous, and increasing functions ina, bsuch that conditions4.3and4.4be satisfied,Γtis positive defined as inTheorem 4.11, andG:a, b → Rbe a function of bounded variation, then for−∞< s /t /0,1/s <∞there existsξ∈I2, withI2

being defined as in4.10, such that

_b

aytτdGτ−

_b

axtτdGτ

_b

aysτdGτ−

_b

axsτdGτ

. 4.25

Proof. Settingfx xt_and_gx_xs_,_{t /}_{s /}₀_,_{1 in}_4.24_{, we get}_4.25_.

Remark 4.16. Since the functionξ→ξt−sis invertible, then from4.25we have

m2≤

⎧ ⎨

⎩ss−

1

tt−1

b

aytτdGτ−

b

axtτdGτ

b

aysτdGτ−

b

axsτdGτ

⎫ ⎬ ⎭

1/t−s

≤M2. 4.26

ξ

f g

−1⎛

⎝

_b

af

yτ dGτ−b

afxτdGτ

_b

ag

yτ dGτ−b

agxτdGτ

⎞

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inequality4.26, we can consider for positive functionsxτandyτsuch that conditions

4.3and4.4are satisfied, and

Mt,s

_Γt

Γs

1/t−s

4.28

Ms,sexp

⎛ ⎝

b

aysτlogyτdGτ−

b

axsτlogxτdGτ

b

aysτdGτ−

b

axsτdGτ

−_ss2s−₋1 1

⎞

⎠_, _{s /}₀_,₁_,

M0,0exp

⎛ ⎜ ⎝

b

alog

2_yτ_dGτ₋b alog

2_xτ_dGτ

2_ablogyτdGτ−_ablogxτdGτ 1

⎞ ⎟

⎠,

M1,1exp

⎛ ⎜ ⎝

_b

ayτlog

2_yτdGτ₋b

axτlog

2_xτdGτ

2_abyτlogyτdGτ−_abxτlogxτdGτ −1

⎞ ⎟

⎠.

4.29

Mt,s≤ Mu,v. 4.30

Proof. SinceΓsis log-convex, therefore by4.22we get4.30.

Remark 4.18. Letx n_i₁piyi/

n

i1pi such thatpi > 0 and

n

i1pi 1. If we substitute in

Theorem 2.6x1;x2;. . .;xn x;x;. . .;x, we get1.2. In fact in such results we have thaty

is monotonicn-tuple. But since the weights are positive, our results are also valid for arbitrary

y.

Acknowledgments

This research work is funded by Higher Education Commission of Pakistan. The research of the third author was supported by the Croatian Ministry of Science, Education and Sports under the Research Grants 117-1170889-0888. The authors thank I. Olkin for many valuable suggestions.

References

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3 A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications, vol. 143 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1979.

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