Sherman’s inequality and its converse for strongly convex functions with applications to generalized f-divergences

Considering the weighted concept of majorization, Sherman obtained generalization of majorization inequalityfor convex functions known as Sherman’s inequality. We extend Sherman’s result to the class of n-strongly convexfunctions using extended idea of convexity to the class of strongly convex functions. We also obtain upper boundfor Sherman’s inequality, called the converse Sherman inequality, and as easy consequences we get Jensen’s as well asmajorization inequality and their conversions for strongly convex functions. Obtained results are stronger versions foranalogous results for convex functions. As applications, we introduced a generalized concept of f -divergence and derivedsome reverse relations for such concept.

PDF

___

[1] Adil Khan M, Ivelić Bradanović S, Pečarić J. Generalizations of Sherman’s inequality by Hermite’s interpolating polynomial. Mathematical Inequalities & Applications 2016; 19 (4): 1181-1192. doi: 10.7153/mia-19-87
[2] Adil Khan M, Ivelić Bradanović S, Pečarić J. Generalizations of Sherman’s inequality by Hermite’s interpolating polynomial and Green function. Konuralp Journal of Mathematics 2016; 4 (2): 255-270.
[3] Agarwal Ravi P, Ivelić Bradanović S, Pečarić J. Generalizations of Sherman’s inequality by Lidstone’s interpolating polynomial. Journal of Inequalities and Applications 2016; 6 (2016): doi: 10.1186/s13660-015-0935-6
[4] Borcea J. Equilibrium points of logarithmic potentials. Transactions of the American Mathematical Society 2007; 359: 3209-3237. doi: 10.1090/S0002-9947-07-04251-1
[5] Burtea AM. Two examples of weighted majorization. Annals of the University of Craiova, Mathematics and Computer Science 2010; 37 (2): 92-99.
[6] Csiszár I. Information-type measures of difference of probability functions and indirect observations. Studia Scientiarum Mathematicarum Hungarica 1967; 2: 299-318.
[7] Csiszár I, Körner J. Information Theory: Coding Theorem for Discrete Memoryless Systems. NY, USA: Academic Press, 1981.
[8] Dragomir SS. Upper and lower bounds for Csiszár f -divergence in terms of the Kullback-Leibler divergence and applications, Inequalities for Csiszár f -Divergence in Information Theory. RGMIA Monographs, Australia: Victoria University, 2000.
[9] Fink AM. Bounds of the deviation of a function from its averages. Czechoslovak Mathematical Journal 1992; 42 (117): 289-310.
[10] Fuchs L. A new proof of an inequality of Hardy-Littlewood-Pólya. Mathematisk Tidsskrift B 1947; (1947): 53-54.
[11] Gera R, Nikodem K. Strongly convex functions of higher order. Advances in Nonlinear Analysis 2011; 74: 661-665.
[12] Ivelić Bradanović S, Latif N, Pečarić J. On an upper bound for Sherman’s inequality. Journal of Inequalities and Applications 2016; (2016): doi: 10.1186/s13660-016-1091-3
[13] Ivelić Bradanović S, Latif N, Pečarić Đ, Pečarić J. Sherman’s and related inequalities with applications in information theory. Journal of Inequalities and Applications 2018; (2018): doi: 10.1186/s13660-018-1692-0
[14] Ivelić Bradanović S, Pečarić J. Extensions and improvements of Sherman’s and related inequalities for n-convex functions. Open Mathematics 2017; 15 (1): doi: 10.1515/math-2017-0077
[15] Ivelić Bradanović S, Pečarić J. Generalizations of Sherman’s inequality. Periodica Mathematica Hungarica 2017; 74 (2): doi: 10.1007/s10998-016-0154-z
[16] Kapur JN. A comparative assessment of various measures of directed divergence. Advances in Management Studies 1984; 3 (1): 1-16. doi: 10.1080/02522667.1983.10698762
[17] Kapur JN. Maximum-Entropy Models in Science and Engineering. NY, USA: John Wiley and Sons, 1989. doi: 10.2307/2532770
18] Kullback S. Information Theory and Statistics. NY, USA: Wiley, 1959.
[19] Lah P, Ribarič M. Converse of Jensen’s inequality for convex functions. Publikacije Elektrotehničkog fakulteta Univerziteta u Beogradu 1973; (412-460): 201-205.
[20] Hardy GH, Littlewood JE, Pólya G. Inequalities. 2nd ed. Cambridge, UK: Cambridge University Press, 1952.
[21] Marshall AW, Olkin I, Barry CA. Inequalities: Theory of Majorization and its Applications. 2nd ed. NY, USA: Springer Series in Statistics, 2011. doi: 10.2307/41305043
[22] Merentes N, Nikodem K. Remarks on strongly convex functions. Aequationes Mathematicae 2010; 80: 193-199. doi: 10.1007/s00010-010-0043-0
[23] Niculescu CP. Choquet theory for signed measures. Mathematical Inequalities & Applications 2002; 5: 479-489. doi: 10.7153/mia-05-47
[24] Niculescu CP, Persson LE. Old and new on the Hermite-Hadamard inequality. Real Analysis Exchange 2004; 29 (2): doi: 10.14321/realanalexch.29.2.0663
[25] Niculescu CP, Rovenţa I. An approach of majorization in spaces with a curved geometry. Journal of Mathematical Analysis and Applications 2014; 411: 119-128. doi: 10.1016/j.jmaa.2013.09.038
[26] Niculescu CP, Rovenţa I. Relative convexity and its applications. Aequationes Mathematicae 2015; 89: 1389-1400. doi: 10.1007/s00010-014-0319-x
[27] Niezgoda M. Nonlinear Sherman-type inequalities. Advances in Nonlinear Analysis 2020; 9 (1): doi: 10.1515/anona- 2018-0098
[28] Niezgoda M. Remarks on Sherman like inequalities for (α, β) -convex functions. Mathematical Inequalities & Applications 2014; 17 (4): 1579-1590. doi: 10.7153/mia-17-116
[29] Niezgoda M. Remarks on convex functions and separable sequences. Discrete Mathematics 2008; 308: 1765-1773. doi: 10.1016/j.disc.2007.04.023
[30] Niezgoda M. Vector joint majorization and generalization of Csiszar-Korner’s inequality for f-divergence. Discrete Applied Mathematics 2016; 198: 195-205. doi: 10.1016/j.dam.2015.06.018
[31] Nikodem K. On Strongly Convex Functions and Related Classes of Functions. Handbook of Functional Equations. NY, USA: Springer, 2014, 365-405. doi: 10.1007/978-1-4939-1246-9_16
[32] Nikodem K, Páles Z. Characterizations of inner product spaces by strongly convex functions. Banach Journal of Mathematical Analysis 2011; 5 (1): 83-87. doi: 10.15352/bjma/1313362982
[33] Pečarić J, Proschan F, Tong YL. Convex functions, Partial Orderings and Statistical Applications. NY, USA: Academic Press, 1992.
[34] Polyak BT. Existence theorems and convergence of minimizing sequences in extremum problems with restrictions. Soviet mathematics-Doklady 1966; 7: 72-75.
[35] Popoviciu T. Les Fonctions Convexes. Paris, France: Hermann et Cie, 1944 (in French).
[36] Shannon CE, Weaver W. The Mathemtiatical Theory of Comnunication. Urbana, USA: University of Illinois Press, 1949.
[37] Sherman S. On a theorem of Hardy, Littlewood, Pólya and Blackwell. Proceedings of the National Academy of Sciences of the United States of America 1957; 37 (1): 826-831. doi: 10.1073/pnas.37.12.826

ISSN: 1300-0098
Yayın Aralığı: Yılda 6 Sayı
Yayıncı: TÜBİTAK

Arşiv