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

1Abdus Salam School of Mathematical Sciences, GC University, 5400 Lahore, Pakistan

2Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, 1000 Zagreb, Croatia

3Faculty of Textile Technology, University of Zagreb, 1000 Zagreb, Croatia

Correspondence should be addressed to M. Adil Khan,[email protected] 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 ∈L¹μis such thata≤ ft ≤bfor all t∈Ω, −∞ ≤a < b≤ ∞,

φ

Ωftdμt

≤

Ωφ ft

dμt 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.

Here and in the whole paper we suppose that all integrals exist. By considering the difference of 1.1for functional in 1Anwar and Peˇcari´c proved an interesting result of log-convexity. We can define this result for integrals as follows.

Theorem 1.2. LetΩ,A, μbe a probability space andf ∈L¹μis such thata≤ ft ≤bfor all t∈Ω, −∞ ≤a < b≤ ∞. Define

Λs

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩ 1 ss−1

Ω

ft_s dμt−

Ωftdμt s

, s /0,1,

log

Ωftdμt

−

Ωlog ft

dμt, s 0,

Ω

ft log

ft dμt−

Ωftdμt

log

Ωftdμt

, s 1,

1.2

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

Λs^u−r ≤Λr^u−sΛu^s−r. 1.3

The following improvement of1.1was obtained in2.

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

Ωφ ft

dμt−φ

Ωftdμt

≥

Ω

φ ft

−φ

fdμt−φ f

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

In fact,Theorem 1.2forΩ a, band 0 < r < s < u, r, s, u /1 was first of all initiated by Simi´c in3.

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 x^r

rr−1w² x^u

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

vx^s/2−1wx^u/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 Λ²_r ≤ΛsΛu. 2.4

Since we have lim_s→0Λs Λ0and lim_s→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 <∞:

Λs^u−r ≤Λr^u−sΛu^s−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

where|aij|kdefine the determinant of orderkwith elementsa_ijandp_ij pip_j/2.

Proof. Consider the function

fx ⁿ

i,j 1

u_iu_j x^pij pij

pij−1 2.7

forx >0 andu_i∈Randp_ij∈I.

So, it holds that

fx ⁿ

i,j 1

uiujx^pij−2 _n

i 1

uix^pi/2−1 ₂

≥0. 2.8

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

i,j 1

u_iu_jΛpij ≥0. 2.9

Therefore,Λpij aijdenote then×nmatrix with elementsa_ijis nonnegative semi definite and2.6is valid forp_ij∈I. Moreover, since we have continuity ofΛpijfor allp_ij,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 1p_ix_iandP_I

i∈Ip_i.

Theorem 3.1. IfΩ,A, μis a probability space and iff∈L¹μis sucha≤ft≤bfort∈Ω,and ifft≥ffort∈Ω⊂Ω(Ωis measurable, i.e.,Ω∈A),−∞< a < b≤ ∞,then

Ωφ ft

dμt−φ

Ωftdμt

≥

Ωsgn

ft−f φ

ft

−φ f

ft

dμt φ

f

−fφ

f

1−2μ Ω,

3.1

where

f

Ωftdμt, 3.2 for monotone convex functionφ : a, b → R. Ifφis monotone concave, then the left-hand side of 3.1should beφ

Ωftdμt−

Ωφftdμt.

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

Ω

φ ft

−φ

fdμt

Ω

φ

ft

−φ f

dμt

Ω\Ω

φ

f

−φ

ft

dμt

Ωφ ft

dμt−

Ω\Ωφ ft

dμt−φ f

μ Ω

φ f

μ Ω\Ω

Ωsgn

ft−f φ

ft

dμt φ f

μ Ω\Ω

−μ Ω

.

3.3

Similarly,

Ω

ft−fdμt

Ωsgn

ft−f

ftdμt f μ

Ω\Ω

−μ Ω

. 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, _n

i 1pi 1.

Ifx_i≥xfori∈I ⊂ {1,2, . . . , n} I_n, then

n i 1

piφxi−φ _n

i 1

pixi

≥

n i 1

p_isgnxi−x

φxi−x_iφx

φx−xφx

1−2P_I .

3.5

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

φ _n

i 1

p_ix_i

−ⁿ

i 1

p_iφxi. 3.6

The following improvement of the Hermite-Hadamard inequality is valid5.

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−a_b

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−a_b

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 for 0< xi < a, pi >01 ≤ i≤n, _n

i 1p_i 1 andP_{k k}

i 1p_i 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 fxⁿ

i 1

pi|xi−x|

,

4.1

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

n i 1

p_if2a−x_i−f2a−x−ⁿ

i 1

p_ifxi fx

≥

n i 1

pisgnxi−x

f2a−xi−fxi xi

f2a−x fx

f2a−x−fx x

f2a−x fx

1−2PI .

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

Letx_i∈0,1/2be such thatx₁≥x₂ ≥ · · · ≥x_k≥x≥x_k1· · · ≥x_n. We denoteG_kandA_k, the weighted geometric and arithmetic means, respectively, that is,

Ak 1 P_k

_k

i 1

pixi

x, Gk

_k

i 1

x_i^pi 1/Pk

, 4.3

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

A_k 1 P_k

k i 1

p_i1−x_i 1−A_k, G_k

_k

i 1

1−x_i^pi 1/Pk

. 4.4

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

G_n G_n ≤ A_n

A_n, 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.

The following improvement of Ky Fan inequality is valid2.

Corollary 4.2. LetAn, GnandA_n, G_nbe as defined earlier. Then, the following inequalities are valid i

A_n/A_n

Gn/G_n ≥exp

n i 1

p_i ln

1−xiAn

xiA_n

− 1 AnA_n

n i 1

p_i|xi−A_n|

, 4.6

ii

A_n/A_n G_n/G_n ≥exp

2Pk

ln

G_kA_n G_kA_n

Ak−An

A_nA_n

ln G_nA_n

A_nG_n

. 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 −ln1−x x

≥

n i 1

pisgnxi−x

ln1−xi

x_i 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∪ {∞}. LetP be 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

is called thef-divergence of the probability distributionsQandP.

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−αs|qs−ps|^αdμs. 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:

I_χ²Q, P

Ω

qs−ps²

ps dμs. 5.4

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

ft t^α, t >0. 5.5

ThenIfgivesα-order entropy

D_αQ, P

Ωq^αsp^1−αsdμs. 5.6 iiiHarmonic distance: let

ft 2t

1t, t >0. 5.7

ThenI_fgives Harmonic distance

DHQ, P

Ω

2psqs

ps qsdμs. 5.8

ivKullback-Leibler: let

ft tlogt, t >0. 5.9

Thenf-divergence functional gives rise to Kullback-Leibler distance10

D_KLQ, 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 g_α : 0,∞ → R,

g_αu

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

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

I_gαQ, P

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

Q−PD_KLP, Q, α 0, αQ−P P−D_αQ, P

α1−α , α∈R\ {0,1}, D_KLQ, 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.

Similar consequence of Theorems 1.2and 2.1in information theory for divergence measures discussed above is the following result.

Theorem 5.3. Letpandqbe positive integrable functions with

Ωpsdμs P,

Ωqsdμs Q. Define the function

Φt

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎩ 1 t1−t

P^tQ^1−t−D_tP, Q

, t /0,1, D_KLQ, P QlogP

Q, t 0,

DKLP, Q PlogQ

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 elementsa_ijandp_ij pip_j/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 M_s,tis defined as

M_s,t Φs

Φt

_1/s−t

, s /t /0,1,

Ms,s exp

P^sQ^1−slogP/Q−D_sP, Q

P^sQ^1−s−DsP, Q − 1−2s s1−s

, s /0,1,

M_0,0 exp Q

logP/Q2−D₀P, Q 2

QlogP/Q−D₀P, Q1

,

5.16

whereD₀P, Q

Ωqslogps/qsdμsandD₀P, Q

Ωqslogps/qs²dμs,

M_1,1 exp Q

logP/Q₂

−D₁P, Q 2

PlogP/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

Mr,t≤Ms,u. 5.18

Proof. By using log convexity of Φt,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 M_s,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 onI^o, then

I_fP, Q−Qf P

Q

≥

I_fP, 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 I^oand let ps/qs> P/Q fors∈Ω⊂Ω

I_fP, 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

Theorem 3.1iiwe get5.22.

Corollary 5.8. It holds that D_HαP, 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 0 one has

D_KLQ, 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

P^αQ^1−α−DαP, Q

α1−α ≥

Ωsgn ps

qs− P Q

αps

qs 1−α−ps^αqs^−α

qsdμs

−α

1−P^α−1Q^1−α

Ωsgn ps

qs− P Q

psdμs

αP Q−αQP Q^1−α−αP/Q

1−P^1−α Q^1−α

1−2Q/Q

α1−α ,

5.27

iiiforα 1 one has

D_KLP, Q P Qlog

P Q

≥

Ωsgn ps

qs− P Q

1−ps

qsps

qslogps 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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