In this paper we give a refinement of Jensen’s integral inequality and its generalization for linear functionals. We also present some applications in Information Theory

Abathun, A

Dragomir, Sever S

Khan, MA

Open Mathematics

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               In this paper we give a refinement of Jensen’s integral inequality and its generalization for linear functionals. We also present some applications in Information Theory.</jats:p

Silvestru Sever Dragomir

Muhammad Adil Khan

Addisalem Abathun

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Refinement of the Jensen integral inequality

Sever Dragomir Silvestru

Adil Khan Muhammad

Abathun Addisalem

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Victoria University Institutional Repository (VUIR)

Dragomir, Silvestru

Name not available

Victoria University Eprints Repository

© 2016 Dragomir et al., published by De Gruyter Open.This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.Open Math. 2016; 14: 221–228Open Mathematics Open AccessResearch ArticleSilvestru Sever Dragomir*, Muhammad Adil Khan, and Addisalem AbathunRefinement of the Jensen integral inequalityDOI 10.1515/math-2016-0020Received September 7, 2015; accepted December 28, 2015.Abstract: In this paper we give a refinement of Jensen’s integral inequality and its generalization for linearfunctionals. We also present some applications in Information Theory.Keywords: Convex functions, Jensen’s inequality, f -divergencesMSC: 26D15, 94A171 IntroductionLet C be a convex subset of the linear space X and f be a convex function on C. If p D .p1; :::; pn/ is probabilitysequence and x D .x1; :::; xn/ 2 Cn, thenf nXiD1pixi!nXiD1pif .xi / (1)is well known in the literature as Jensen’s inequality.The Lebesgue integral version of the Jensen inequality is given below:Theorem 1.1. Let .;ƒ;/ be a measure space with 0 < ./ < 1 and let  W I ! R be a convex functiondefined on an open interval I in R. If f W ! I is such that f;  ı f 2 L.;ƒ;/, then0@ 1./Zfd1A  1./Z.f /d: (2)In case when  is strictly convex on I one has equality in .2/ if and only if f is constant almost everywhere on .The Jensen inequality for convex functions plays a crucial role in the Theory of Inequalities due to the fact that otherinequalities such as the arithmetic mean-geometric mean inequality, the Hölder and Minkowski inequalities, the KyFan inequality etc. can be obtained as particular cases of it.There is an extensive literature devoted to Jensen’s inequality concerning different generalizations, refinements,counterparts and converse results, see, for example [1–9].In this paper we give a refinement of Jensen’s integral inequality and its generalization for linear functionals.We also present some applications in Information Theory for example for Kullback-Leibler, total variation and KarlPearson 2-divergences etc.*Corresponding Author: Silvestru Sever Dragomir: School of Computer Science and Mathematics, Victoria University of Technol-ogy PO Box 14428, MCMC 8001 Victoria, Australia, E-mail: sever.dragomir@vu.edu.auMuhammad Adil Khan: Department of Mathematics, University of Peshawar, Peshawar, Pakistan, E-mail: adilswati@gmail.comAddisalem Abathun: Department of Mathematics, Adis Ababa University, PO Box 1176, Ethiopia,E-mail: addisalem.abathun@aau.edu.etBrought to you by | Victoria University AustraliaAuthenticatedDownload Date | 5/15/17 5:43 AM222 Silvestru Sever Dragomir et al.2 Main resultsLet .;ƒ;/ be a measure space with 0 < ./ <1 and L.;ƒ;/ D ff W ! R W f is  measurable andRf .t/d.t/ <1g be a Lebesgue space. Consider the set S D f! 2 ƒ W .!/ ¤ 0 and .!/ D . n !/ ¤ 0gand  W .a; b/ ! R be a convex function defined on an open interval .a; b/. If f W  ! .a; b/ is such thatf;  ı f 2 L.;ƒ;/, then for any set ! 2 S, define the functional as±.; f I!/ D .!/./0@ 1.!/Z!fd1AC .!/./0@ 1.!/Z!fd1A : (3)We give the following refinement of Jensen’s inequality.Theorem 2.1. Let .;ƒ;/ be a measure space with 0 < ./ <1 and let  W .a; b/! R be a convex functiondefined on an open interval .a; b/. If f W ! .a; b/ is such that f;  ı f 2 L.;ƒ;/, then for any set ! 2 S wehave0@ 1./Zfd1A  ±.; f I!/  1./Z.f /d: (4)Proof. As for any ! 2 S we have0@ 1./Zfd1A D  24.!/./0@ 1.!/Z!fd1AC .!/./0@ 1.!/Z!fd1A35 :Therefore by the convexity of the function  we get0@ 1./Zfd1A  .!/./0@ 1.!/Z!fd1AC .!/./0@ 1.!/Z!fd1A D ±.; f I!/; (5)Also for any ! 2 S and by the Jensen inequality we have±.; f I!/ D .!/./0@ 1.!/Z!fd1AC .!/./0@ 1.!/Z!fd1A 1./Z!.f /dC 1./Z!.f /dD 1./Z.f /d: (6)From (5) and (6) we have (4).Remark 2.2. We observe that the inequality (4) can be written in an equivalent form asinf!2S±.; f I!/  0@ 1./Zfd1Aand1./Z.f /d  sup!2S±.; f I!/:Remark 2.3. If ;;  2 S and if we take ! D ; or ! D , then we have ±.; f I!/ is equal to the left hand sideof (2). In this case (5) holds trivially.Brought to you by | Victoria University AustraliaAuthenticatedDownload Date | 5/15/17 5:43 AMRefinement of the Jensen integral inequality 223Particularly Riemann integral version can be given as:Corollary 2.4. Let  W Œa; b ! R be a convex function defined on the interval Œa; b. If f W Œc; d  ! Œa; b; p WŒc; d ! RC are such that f; fp and . ı f /p are all integrable on Œc; d , then we haveinfx2.c;d/24x   cd   c 0@ 1x   cxZcp.t/f .t/dt1AC d   xd   c 0@ 1d   xdZxp.t/f .t/dt1A35  0@ 1d   cdZcp.t/f .t/dt1A ;1d   cdZcp.t/.f .t//dt  supx2Œc;d24x   cd   c 0@ 1x   cxZcp.t/f .t/dt1AC d   xd   c 0@ 1d   xdZxp.t/f .t/dt1A35 :As a simple consequence of Theorem 2.1 we can obtain refinement of Hermite-Hadamard inequality:Corollary 2.5. If  W Œa; b ! R is a convex function defined on the interval Œa; b, then for any Œc; d   Œa; b wehaved C c2 infx2Œc;dx   cd   c x C c2C d   xd   c d C x2;1d   cdZc.t/dt  supx2Œc;dx   cd   c x C c2C d   xd   c d C x2:3 Further generalizationLetE be a nonempty set,A be an algebra of subsets ofE, andL be a linear class of real-valued functions f W E ! Rhaving the properties:L1 : f; g 2 L) . f˛ C ˇg/ 2 L for all ˛; ˇ 2 R;L2 : 1 2 L, i.e., if f .t/ D 1 for all t 2 E, then f 2 L;L3 : f 2 L, E1 2 A) f:E1 2 L,where E1 is the indicator function of E1. It follows from L2; L3 that E1 2 L for every E1 2 A.A positive isotonic linear functional A W L! R is a functional satisfying the following properties:A1 : A. f˛ C ˇg/ D ˛A.f /C ˇA.g/ for f; g 2 L; ˛; ˇ 2 R;A2 : f 2 L; f .t/  0 on E ) A.f /  0;It follows from L3 that for every E1 2 A such that A.E1/ > 0; the functional AE1 is defined for a fixed positiveisotonic linear functional A as AE1.f / D A.f:E1 /A.E1 / , for all f 2 L, with A.1/ D 1. Furthermore, we observe thatA.E1/C A.EnE1/ D 1;A.f / D A.f:E1/C A.f:EnE1/: (7)Jessen (see [10, p-47]) gave the following generalization of Jensen’s inequality for convex functions.Theorem 3.1. Let L satisfy L1 and L2 on a nonempty set E, and assume that  W Œa; b ! R be a continuousconvex function. If A is linear positive functional with A.1/ D 1, then for all f 2 L such that .f / 2 L we haveA.f / 2 Œa; b and.A.f //  A..f //I (8)Brought to you by | Victoria University AustraliaAuthenticatedDownload Date | 5/15/17 5:43 AM224 Silvestru Sever Dragomir et al.The following refinement of (8) holds.Theorem 3.2. Under the above assumptions, if  W Œa; b! R is a continuous convex function, then.A.f //  D.A; f; IE1/  A..f //I (9)whereD.A; f; IE1/ D A.E1/A.f:E1/A.E1/C A.EnE1/A.f:EnE1/A.EnE1/for all non empty set E1 2 A such that 0 < A.E1/ < 1Proof. SinceD.A; f; IE1/ D A.E1/A.f:E1 /A.E1 /C A.EnE1/A.f:EnE1 /A.EnE1 /i. e. D.A; f; IE1/ D A.E1/ .AE1.f //C A.EnE1/ AEnE1.f /:Using the inequality (8) we obtainD.A; f; IE1/  A. .f / :E1/C A  .f / :EnE1 D A. .f //: (10)This proves the second inequality in (9).The first inequality follows by using definition of convex function and identity (7).4 Applications for Csiszár divergence measuresLet .;ƒ;/ be a probability measure space. Consider the set of all density functions on  to be S WD fpjp W !R; p.s/ > 0;Rp.s/d.s/ D 1g.Csiszár introduced the concept of f  divergence for a convex function f W .0;1/ ! . 1;1/ (cf. [11] , seealso [12]) byIf .q; p/ DZp.s/ fq.s/p.s/d.s/; p; q 2 S:By appropriately defining the convex function f , various divergences can be derived. We give some importantf -divergences, playing a significant role in Information Theory and Statistics.(i) The class of -divergences: The f -divergences, in this class, are generated by the family of functionsf˛.u/ D j u   1 j˛ u  0 and ˛  1:If˛ .q; p/ DZp1 ˛.s/ j q.s/   p.s/ j˛ d.s/:For ˛ D 1; it gives the total variation distance.V.q; p/ DZj q.s/   p.s/ j d.s/:For ˛ D 2; it gives the Karl Pearson 2-divergence,I2.q; p/ DZŒq.s/   p.s/2p.s/d.s/:(ii) ˛-order Renyi entropy : For ˛ > 1 letf .t/ D t˛; t > 0:Brought to you by | Victoria University AustraliaAuthenticatedDownload Date | 5/15/17 5:43 AMRefinement of the Jensen integral inequality 225Then If gives ˛-order entropyD˛.q; p/ DZq˛.s/ p1 ˛.s/ d.s/:(iii) Harmonic distance: Letf .t/ D   2t1C t ; t > 0:Then If gives Harmonic distanceDH .q; p/ DZ2p.s/ q.s/p.s/C q.s/ d.s/:(iv) Kullback-Leibler: Letf .t/ D t log t; t > 0:Then f -divergence functional give rise to Kullback-Leibler distance [13]DKL.q; p/ DZq.s/ logq.s/p.s/d.s/:One parametric generalization of the Kullback-Leibler [13] relative information was studied in a different way byCressie and Read [14].(v) Jeffreys divergence: Letf .t/ D .t   1/ log t; t > 0:Then f -divergence functional give Jeffreys divergenceJ.q; p/ DZ!.p.s/   q.s// lnq.s/p.s/d.s/:(vi) The Dichotomy class: This class is generated by the family of functions g˛ W .0;1/! R;g˛.u/ D8ˆˆˆˆ<ˆˆˆˆˆˆ:u   1   log u; ˛ D 01˛.1 ˛/ Œ ˛ uC 1   ˛   u˛ ; ˛ 2 Rnf0; 1g;1   uC u log u; ˛ D 1.(11)This class gives, for particular values of ˛; some important divergences. For instance, for ˛ D 12it providesa distance, namely, the Hellinger distance.There are various other divergences in Information Theory and Statistics such as Arimoto-type divergences,Matushita’s divergence, Puri-Vincze divergences etc. ( cf. [15], [16]) used in various problems in Information Theoryand statistics. An application of Theorem 1.1 is the following result given by Csiszár and Korner (cf. [17]).Theorem 4.1. Let f W Œ0;1/! R be a convex function and p; q be positive functions from S . Then the followinginequality is valid,If .q; p/  f .1/: (12)Theorem 4.2. Let f W Œ0;1! R be a convex function, then for any p and q in S we have:If .q; p/  .!/f0@ 1.!/Z!q.s/d.s/1AC .!/f 0@ 1.!/Z!q.s/d.s/1A  f .1/: (13)Proof. By substituting .s/ D f .s/, f .s/ D q.s/p.s/and d.s/ D p.s/d.s/ in Theorem 2.1, we deduce (13).Brought to you by | Victoria University AustraliaAuthenticatedDownload Date | 5/15/17 5:43 AM226 Silvestru Sever Dragomir et al.Proposition 4.3. Let p; q 2 S , then we haveV.q; p/  2 sup!2SˇˇˇˇˇˇZ!q.s/d.s/   .!/ˇˇˇˇˇˇ . 0/: (14)Proof. By putting f .x/ D jx   1j for all x  0 in Theorem 4.2 we get (14).Proposition 4.4. For any p, q 2 S ,I2.q; p/  sup!2S8<:R!q.s/d.s/   .!/2.!/.1   .!//9=;  4 sup!2S8<:Z!q.s/d.s/   .!/29=; . 0/: (15)Proof. By making use of the function f .x/ D .t   1/2 in Theorem 4.2 we getZp.s/q.s/p.s/  12d.s/  sup!2S8ˆ<ˆ:.!/0@ 1.!/Z!q.s/d.s/   11A2C.!/0@ 1.!/Z!q.s/d.s/   11A29>=>; . 0/i.e.Z.q.s/   p.s//2p.s/d.s/  sup!2S( R!q.s/d.s/   .!/2.!/.1   .!//). 0/:Since by Arithmetic-Geometric mean inequality we have.!/.1   .!//  14h.!/C .1   .!//i2 D 14;therefore  R!q.s/d.s/   .!/2.!/.1   .!/  4 Z!q.s/d.s/   .!/2. 0/:Proposition 4.5. For any p, q 2 S , we have:DKL.q; p/  ln"1   R!q.s/d.s/1   .!/1 R! q.s/d.s/:R!q.s/d.s/.!/R! q.s/d.s/#. 0/: (16)Proof. By putting f .t/ D t ln.t/ in Theorem 4.2 one can get first inequality in (16).To prove the second inequality, we utilize the inequality between the geometric mean and harmonic mean,x˛y1 ˛  1˛xC 1 ˛y; x; y; ˛ 2 Œ0; 1;we have forx DR!q.s/d.s/.!/; y D 1  R!q.s/d.s/1   .!/ and ˛ DZ!q.s/d.s/ that1   R!q.s/d.s/1   .!/1 R! q.s/d.s/:R!q.s/d.s/.!/R! q.s/d.s/ 1;for any ! 2 S; which implies the second inequality in (16).Brought to you by | Victoria University AustraliaAuthenticatedDownload Date | 5/15/17 5:43 AMRefinement of the Jensen integral inequality 227Proposition 4.6. For any p; q 2 S; we have:J.q; p/  ln sup!2S(.1   R!q.s/d.s//.!/.1   .!// R!q.s/d.s/..!/ R! q.s/d.s//)! sup!2S ..!/   R!q.s/d.s//2R!q.s/d.s/C .!/   2 R!q.s/d.s/.!/! 0: (17)Proof. By putting f .x/ D .x   1/ ln.x/; x > 0 in Theorem 4.2 we haveZ!p.s/q.s/p.s/  1lnq.s/p.s/d.s/  sup!2S.!/R!q.s/d.s/.!/  1lnR!q.s/d.s/.!/C. N!/RN! q.s/d.s/. N!/   1lnRN! q.s/d.s/. N!/D sup!2S0@0@Z!q.s/d.s/   .!/1A lnR! q.s/d.s/.!/C0@ZN!q.s/d.s/   . N!/1A lnR N! q.s/d.s/. N!/1Athat isJ.q; p/  sup!2S0@0@.!/   Z!q.s/d.s/1A ln1   R! q.s/d.s/1   .!/ 0@.!/   Z!q.s/d.s/1A lnR! q.s/d.s/.!/1Aproving the first inequality in (17).Utilizing the elementary inequality for positive numbers ,ln b   ln ab   a 2aC b ; a; b > 0we have0@.!/   Z!q.s/d.s/1Aln1   R! q.s/d.s/1   .!/  lnR!q.s/d.s/.!/D0@.!/   Z!q.s/d.s/1A ln 1 R! q.s/d.s/1 .!/    ln  R! q.s/d.s/.!/ 1 R! q.s/d.s/1 .!/  R! q.s/d.s/.!/1   R!q.s/d.s/1   .!/  R!q.s/d.s/.!/D .!/   R!q.s/d.s/2.!/.1   .!// :ln1 R! q.s/d.s/1 .!/  ln R! q.s/d.s/.!/1 R! q.s/d.s/1 .!/  R! q.s/d.s/.!/ .!/   R!q.s/d.s/2.!/.1   .!// :21 R! q.s/d.s/1 .!/ CR! q.s/d.s/.!/D 2..!/  R!q.s/d.s//2R!q.s/d.s/C .!/   2 R!q.s/d.s/.!/ 0;for each ! 2 S; giving the second inequality in (17).Brought to you by | Victoria University AustraliaAuthenticatedDownload Date | 5/15/17 5:43 AM228 Silvestru Sever Dragomir et al.Proposition 4.7. For any p, q 2 S , we have:D˛.q; p/  sup!2S24..!//1 ˛ 0@Z!q.s/d.s/1A˛ C .1   .!//1 ˛ 0@1   Z!q.s/d.s/1A˛35  1: (18)Proof. By putting f .x/ D x˛ for ˛ > 1; x > 0; in Theorem 4.2 we get the required inequalities.Acknowledgement: The authors express their sincere thanks to the referees for their careful reading of themanuscript and very helpful suggestions that improved the manuscript.References[1] Adil Khan M., Anwar M., Jakšetic´ J., and Pecˇaric´ J., On some improvements of the Jensen inequality with some applications,J. Inequal. Appl. 2009 (2009), Article ID 323615, 15 pages.[2] Adil Khan M., Khan G. A., Ali T., Batbold T., and Kilicman A., Further refinement of Jensen’s type inequalities for the functiondefined on the rectangle, Abstr. Appl. Anal. 2013 (2013), Article ID 214123, 1-8.[3] Adil Khan M., Khan G. A., Ali T., and Kilicman A., On the refinement of Jensen’s inequality, Appl. Math. Comput. 262(1) (2015),128-135.[4] Beesack P. 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