On Some Improvements of the Jensen Inequality with Some Applications

Volume 2009, Article ID 323615,15pages doi:10.1155/2009/323615

Research Article

On Some Improvements of the Jensen

Inequality with Some Applications

M. Adil Khan,

_{M. Anwar,}

_{J. Jak ˇseti ´c,}

_{and J. Pe ˇcari ´c}

1_{Abdus Salam School of Mathematical Sciences, GC University, 5400 Lahore, Pakistan} 2_{Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb,}

1000 Zagreb, Croatia

3_{Faculty of Textile Technology, University of Zagreb, 1000 Zagreb, Croatia}

Correspondence should be addressed to M. Adil Khan,adilbandai@yahoo.com

Received 23 April 2009; Accepted 10 August 2009

Recommended by Sever Silvestru Dragomir

An improvement of the Jensen inequality for convex and monotone function is given as well as various applications for mean. Similar results for related inequalities of the Jensen type are also obtained. Also some applications of the Cauchy mean and the Jensen inequality are discussed.

Copyrightq2009 M. Adil Khan 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.

1. Introduction

The well-known Jensen’s inequality for convex function is given as follows.

Theorem 1.1. IfΩ,A, μis a probability space and iff ∈L1_{μis such thata≤ ft ≤bfor all}

t∈Ω, −∞ ≤a < b≤ ∞,

φ

Ωftdμt

≤

Ωφ

ftdμt 1.1

is valid for any convex functionφ :a, b → R. In the case whenφis strictly convex ona, bone has equality in1.1if and only iffis constant almost everywhere onΩ.

Theorem 1.2. LetΩ,A, μbe a probability space andf ∈L1_{μis such thata≤ ft ≤bfor all}

t∈Ω, −∞ ≤a < b≤ ∞. Define

Λs

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

1

ss−1

Ω

ftsdμt−

Ωftdμt

s

, s /0,1,

log

Ωftdμt

−

Ωlog

ftdμt, s 0,

Ω

ftlogftdμt−

Ωftdμt

log

Ωftdμt

, s 1,

1.2

and letΛsbe positive. ThenΛsis log-convex, that is, for−∞< r < s < u <∞,the following is valid

Λsu−r _≤Λru−s_Λus−r_{. 1.3}

The following improvement of1.1was obtained in2.

Theorem 1.3. Let the conditions ofTheorem 1.1be fulfilled. Then

Ωφ

ftdμt−φ

Ωftdμt

≥

Ω

φft−φfdμt−φf

Ω

ft−fdμt,

1.4

whereφxrepresents the right-hand derivative ofφand

f

Ωftdμt. 1.5

Ifφis concave, then left-hand side of 1.4should beφ_Ωftdμt−_Ωφftdμt.

In this paper, we give another proof and extension of Theorem 1.2 as well as improvements ofTheorem 1.3for monotone convex function with some applications. Also we give applications of the Jensen inequality for divergence measures in information theory and related Cauchy means.

2. Another Proof and Extension of

Theorem 1.2

Moreover, in his proof, he has used convex functions defined onI −∞,0∪0,1∪

1,∞ see3, Theorem 1. In his proof, he has used the following function:

λx v2 x

s

ss−12vw

xr

rr−1w 2 xu

uu−1, 2.1

wherer su/2 andv, w, r, s, uare real withr, s, t∈I.

In1we have given correct proof by using extension of2.1, so that it is defined on R.

Moreover, we can give another proof so that we use only 2.1 but without using convexity as in3.

Proof ofTheorem 1.2. Consider the functionλxdefined, as in3, by2.1. Now

λx vxs/2−1wxu/2−12≥0, forx >0, 2.2

that is,λxis convex. By using1.1we get

v2Λs2vwΛrw2Λu≥0. 2.3

Therefore,2.3is valid for alls, r, u∈I. Now since left-hand side of2.3is quadratic form, by the nonnegativity of it, one has

Λ2

su/2 Λ2r ≤ΛsΛu. 2.4

Since we have lims→0Λs Λ0and lims→1Λs Λ1, we also have that2.4is valid forr, s, u∈

R. Sos →Λsis log-convex function in the Jensen sense onR.

Moreover, continuity of Λs implies log-convexity, that is, the following is valid for

−∞< r < s < u <∞:

Λsu−r _≤Λru−s_Λus−r_{. 2.5}

Let us note that it was used in4to get corresponding Cauchy’s means. Moreover, we can extend the above result.

Theorem 2.1. Let the conditions of Theorem 1.2 be fulfilled and let pi i 1,2, . . . , n be real

numbers. Then

Λpij

k≥0 k 1,2, . . . , n, 2.6

Proof. Consider the function

fx

n

i,j 1

uiuj

xpij

pij

pij−1

2.7

forx >0 andui∈Randpij∈I.

So, it holds that

fx

n

i,j 1

uiujxpij−2

_n

i 1

uixpi/2−1

2

≥0. 2.8

Sofxis convex function, and as a consequence of1.1, one has

n

i,j 1

uiujΛpij ≥0. 2.9

Therefore,Λp_ij aijdenote then×nmatrix with elementsaijis nonnegative semi definite and2.6is valid forpij∈I. Moreover, since we have continuity ofΛpijfor allpij,2.6is valid for allpi∈Ri 1,2, . . . , n.

Remark 2.2. InTheorem 2.1, if we setn 2,we getTheorem 1.2.

3. Improvements of the Jensen Inequality for Monotone

Convex Function

In this section and in the following section, we denotex n_{i 1}pixiandPI

i∈Ipi.

Theorem 3.1. IfΩ,A, μis a probability space and iff∈L1_{μis sucha≤ft≤bfort∈Ω,and} ifft≥ffort∈Ω⊂Ω(Ωis measurable, i.e.,Ω∈A),−∞< a < b≤ ∞,then

Ωφ

ftdμt−φ

Ωftdμt

≥

Ωsgn

ft−fφft−φfftdμt φf−fφf1−2μΩ,

3.1

where

f

Ωftdμt, 3.2

for monotone convex functionφ : a, b → R. Ifφis monotone concave, then the left-hand side of

Proof. Consider the case whenφis nondecreasing ona, b. Then

Ω

φft−φfdμt

Ω

φft−φfdμt

Ω\Ω

φf−φftdμt

Ωφ

ftdμt−

Ω\Ωφ

ftdμt−φfμΩφfμΩ\Ω

Ωsgn

ft−fφftdμt φfμΩ\Ω−μΩ.

3.3

Similarly,

Ω

ft−fdμt

Ωsgn

ft−fftdμt fμΩ\Ω−μΩ. 3.4

Now from1.4,3.3, and3.4we get3.1.

The case whenφis nonincreasing can be treated in a similar way.

Of course a discrete inequality is a simple consequence ofTheorem 3.1.

Theorem 3.2. Letφ:a, b → Rbe a monotone convex function,xi∈a, b, pi >0, ni 1pi 1.

Ifxi≥xfori∈I ⊂ {1,2, . . . , n} In, then

n

i 1

piφxi−φ

_n

i 1

pixi

≥n

i 1

pisgnxi−x

φxi−xiφxφx−xφx1−2PI

.

3.5

Ifφis monotone concave, then the left-hand side of 3.5should be

φ _n

i 1

pixi

−n

i 1

piφxi. 3.6

Corollary 3.3. Letφ:a, b → Rbe a differentiable convex. Then

ithe inequality

1

b−a b

a

φtdt−φ

ab

2

≥ 1

b−a b

a

φt−φ

ab

2

dt

−

b−a

4

φ

ab

2

3.7

holds.

If φ is differentiable concave, then the left-hand side of 3.7 should beφab/2−

1/b−ab_aφtdt;

iiif φ is monotone, then the inequality

1

b−a _b

a

φtdt−φ

ab

2

≥ 1 b−a

_b

a

sgn

t−ab

2

φt−tφ

ab

2

dt

3.8

holds. Ifφis differentiable and monotone concave then the left-hand side of 3.8should be

φab/2−1/b−ab_aφtdt.

Proof. iSettingΩ a, b, ft t, dμt dt/b−ain1.4, we get3.7.

iiSettingft t, dμt dt/b−a, andΩ a, bin3.1, we get3.8.

4. Improvements of the Levinson Inequality

Theorem 4.1. If the third derivative offexist and is nonnegative, then for0< xi < a, pi >01 ≤

i≤n, n_{i 1}pi 1andPk

_k

i 1pi 2≤k≤n−1one has i

n

i 1

pif2a−xi−f2a−x− n

i 1

pifxi fx

≥n

i 1

pif2a−xi−fxi−f2a−x fx

−f2a−x fx

n

i 1

pi|xi−x|

,

iiifφx f2a−x−fxis monotone andxi≥xfori∈I⊂ {1,2, . . . , n} In, then

n

i 1

pif2a−xi−f2a−x− n

i 1

pifxi fx

≥n

i 1

pisgnxi−x

f2a−xi−fxi xi

f2a−x fx

f2a−x−fx xf2a−x fx1−2PI

.

4.2

Proof. iAs for 3-convex functionf :0,2a → Rthe functionφx f2a−x−fxis convex on0, a, so by settingφ f2a−x−fxin the discrete case of2, Theorem 2, we get4.1.

iiAsf2a−x−fxis monotone convex, so by settingφ f2a−x−fxin3.5, we get5.16.

Ky Fan Inequality

Letxi∈0,1/2be such thatx1≥x2 ≥ · · · ≥xk≥x≥xk1· · · ≥xn. We denoteGkandAk, the

weighted geometric and arithmetic means, respectively, that is,

Ak 1

Pk

_k

i 1

pixi

x, Gk

_k

i 1

xpi

i

1/Pk

, 4.3

and also byA_kandG_k, the arithmetic and geometric means of 1−xi,respectively, that is,

A_k 1 Pk

k

i 1

pi1−xi 1−Ak, G_k

_k

i 1

1−xipi

1/Pk

. 4.4

The following remarkable inequality, due to Ky Fan, is valid6, page 5,

Gn

Gn

≤ An

An

, 4.5

with equality sign if and only ifx1 x2 · · · xn.

Inequality4.5has evoked the interest of several mathematicians and in numerous articles new proofs, extensions, refinements and various related results have been published

7.

Corollary 4.2. LetAn, GnandAn, Gnbe as defined earlier. Then, the following inequalities are valid i

An/An

Gn/Gn

≥exp

n

i 1

pi

ln

1−xiAn

xiAn

− 1

AnAn n

i 1

pi|xi−An|

, 4.6

ii

An/An

Gn/Gn ≥

exp

2Pk

ln

G_kAn

GkAn

Ak−An

AnAn

ln

_G

nAn

AnGn

. 4.7

Proof. iSettinga 1/2,fx lnxin4.1, we get4.6.

iiConsidera 1/2 andfx lnx,thenφx ln1−x−lnxis strictly monotone convex on the interval0,1/2and has derivative

φx − 1

xx−1. 4.8

Then the application of inequality4.2to this function is given by

n

i 1

piln1−xi

xi −ln

1−x x

≥n

i 1

pisgnxi−x

ln1−xi

xi

xi

x1−x

ln1−x

x

1 1−x

1−2Pk.

4.9

From4.9we get4.7.

5. On Some Inequalities for Csisz ´ar Divergence Measures

LetΩ,A, μbe a measure space satisfying|A|>2 andμaσ-finite measure onΩwith values inR∪ {∞}. LetPbe the set of all probability measures on the measurable spaceΩ,Awhich are absolutely continuous with respect toμ. ForP, Q ∈ P, let p dP/dμand q dQ/dμ

denote the Radon-Nikodym derivatives ofPandQwith respect toμ,respectively.

Csisz´ar introduced the concept off-divergence for a convex function, f : 0,∞ →

−∞,∞that is continuous at 0 as followscf.8, see also9.

Definition 5.1. LetP, Q∈P. Then

IfQ, P

Ωpsf

qs ps

dμs, 5.1

We give some important f-divergences, playing a significant role in Information Theory and statistics.

iThe class of χ-divergences: thef-divergences, in this class, are generated by the family of functions:

fαu |u−1|α u≥0, α≥1,

IfαQ, P

Ωp

1−α_s|qs−ps|α_dμs. 5.2

Forα 1, it gives the total variation distance:

VQ, P

Ω

qs−psdμs. 5.3

Forα 2, it gives the Karl pearsonχ2_-divergence:

Iχ2Q, P

Ω

qs−ps2

ps dμs. 5.4

iiTheα-order Renyi entropy: forα∈R\ {0,1}, let

ft tα, t >0. 5.5

ThenIfgivesα-order entropy

DαQ, P

Ωq

α_sp1−α_{sdμs. 5.6}

iiiHarmonic distance: let

ft 2t

1t, t >0. 5.7

ThenIfgives Harmonic distance

DHQ, P

Ω

2psqs

ps qsdμs. 5.8

ivKullback-Leibler: let

Thenf-divergence functional gives rise to Kullback-Leibler distance10

DKLQ, P

Ωqslog

qs ps

dμs. 5.10

The one parametric generalization of the Kullback-Leibler10relative information studied in a different way by Cressie and Read11.

v The Dichotomy class: this class is generated by the family of functions gα : 0,∞ → R,

gαu ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

u−1−logu, α 0,

1

α1−ααu1−α−u

α_{, α∈R\ {0,1},}

1−uulogu, α 1.

5.11

This class gives, for particular values of α, some important divergences. For instance, for

α 1/2,we have Hellinger distance and some other divergences for this class are given by

IgαQ, P

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

Q−PDKLP, Q, α 0,

αQ−P P−DαQ, P

α1−α , α∈R\ {0,1}, DKLQ, P P−Q, α 1,

5.12

wherepxandqxare positive integrable functions with_Ωpsdμs P, _Ωqsdμs Q.

There are various other divergences in Information Theory and statistics such as Arimoto-type divergences, Matushita’s divergence, Puri-Vincze divergences cf. 12–14

used in various problems in Information Theory and statistics. An application ofTheorem 1.1

is the following result given by Csisz´ar and K ¨ornercf.15.

Theorem 5.2. Letf :0,∞ → Rbe convex, and letpandqbe positive integrable function with

Ωpsdμs P,

Ωqsdμs Q. Then the following inequality is valid:

IfP, Q≥Qf

P Q

, 5.13

whereIfP, Q

Ωqsfps/qsdμs.

Proof. By substitutingφs fs, fs ps/qsanddμs qsdμsinTheorem 1.1

we get5.13.

Theorem 5.3. Letpandqbe positive integrable functions with_Ωpsdμs P, _Ωqsdμs Q. Define the function

Φt

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

1

t1−t

Pt_Q1−t_{−DtP, Q, t /0,1,}

DKLQ, P Qlog

P

Q, t 0,

DKLP, Q Plog

Q

P, t 1,

5.14

and letΦtbe positive. Then

iit holds that

Φpij

k≥0 k 1,2, . . . , n, 5.15

where|aij|kdefine the determinant of ordernwith elementsaijandpij pipj/2, ii Φtis log-convex.

As we said in 4 we define new means of the Cauchy type, here we define an application of these means for divergence measures in the following definition.

Definition 5.4. Let p and q be positive integrable functions with _Ωpsdμs P, _Ωqsdμs Q. The mean Ms,tis defined as

Ms,t

_Φs

Φt

1/s−t

, s /t /0,1,

Ms,s exp

Ps_Q1−s_logP/Q−D

sP, Q

Ps_Q1−s_{−DsP, Q} −

1−2s s1−s

, s /0,1,

M0,0 exp

QlogP/Q2−D₀P, Q

2QlogP/Q−D₀P, Q1

,

5.16

whereD₀P, Q _Ωqslogps/qsdμsandD₀P, Q _Ωqslogps/qs2dμs,

M1,1 exp

QlogP/Q2−D₁P, Q

2PlogP/Q−D₁P, Q−1

, 5.17

whereD₁P, Q _Ωpslogqs/psdμsandD₁P, Q _Ωpslogqs/ps2dμs.

Theorem 5.5. Letr, s, t, ube nonnegative reals such thatr ≤t, s≤u,then

Proof. By using log convexity ofΦt,we get the following result forr, s, t, u ∈ R such that

r≤t, s≤uandr /s, t /u

_Φs

Φr

1/s−r

≤ _Φu

Φt

1/u−t

. 5.19

Also forr s, t u,we consider limiting case and the result follows from continuity of Ms,u.

An application ofTheorem 1.3in divergence measure is the following result given in

16.

Theorem 5.6. Letf :I ⊆R → Rbe differentiable convex function onIo, then

IfP, Q−Qf

P Q

≥IfP, Q−Qf

P Q

−fP/Q

Q Q

, 5.20

where

Q

Ω

_{Qps−P qsdμs. 5.21}

Proof. By substitutingφs fs, fs ps/qs,anddμs →qsdμsinTheorem 1.3, we get5.20.

Theorem 5.7. Let f : I ⊆ R → R be differentiable monotone convex function on Ioand let

ps/qs> P/Q fors∈Ω⊂Ω

IfP, Q−Qf

P Q

≥

Ωsgn

_ps

qs− P Q

f

_ps

qs

qsdμs

−f

P Q

Ωsgn

_ps

qs− P Q

psdμs

Q

f

P Q

− P

Q f

P

Q

1 − 2Q

Q

,

5.22

where

Q

Ωqsdμs, 5.23

andΩas inTheorem 5.7.

Proof. By substituting φs fs, fs ps/qs and dμs → qsdμs in

Corollary 5.8. It holds that

DHαP, Q− 2P Q PQ ≥

Ωsgn

ps qs −

P Q

2psqs ps qsdμs

− 2Q2

PQ2

Ωsgn

_ps

qs− P Q

psdμs

2P Q PQ−

2P Q2

PQ2

1−2Q

Q , 5.24 where Q

Ωqsdμs, 5.25

andΩas inTheorem 5.7.

Proof. The proof follows by settingft 2t/1t, t >0 inTheorem 5.7.

Corollary 5.9. Letgα:R → Rbe as given in5.11, then

iforα 0one has

DKLQ, P Qlog

P Q ≥ Ωsgn ps qs−

P Q

ps

qs−1−log

ps qs

qsdμs

−P−Q P

Ωsgn

_ps

qs− P Q

psdμs Q log

P Q

1− 2Q

Q

,

5.26

iiforα∈R\ {0,1}one has

Pα_Q1−α_{−DαP, Q}

α1−α ≥

Ωsgn

_ps

qs− P Q

αps

qs 1−α−ps

α_qs−α_qsdμs

−α1−Pα−1_Q1−α

Ωsgn

ps qs−

P Q

psdμs

αP Q−αQP Q1−α−αP/Q1−P1−α Q1−α1−2Q/Q

α1−α ,

iiiforα 1one has

DKLP, Q

P Qlog

P Q

≥

Ωsgn

_ps

qs− P Q

1−ps

qs ps qslog

ps qs

qsdμs

−log

P Q

Ωsgn

_ps

qs − P Q

psdμs

Q−P

1−2Q

Q

,

5.28

where

Q

Ωqsdμs, 5.29

and Ωas inTheorem 5.7.

Proof. The proof follows be settingf gαto be as given in5.11, inTheorem 3.1.

Acknowledgments

This research work is funded by the Higher Education Commission Pakistan. The research of the fourth author is supported by the Croatian Ministry of Science, Education and Sports under the Research Grants 117-1170889-0888.

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