Refinement of Jensen’s inequality and estimation of f- and Rényi divergence via Montgomery identity

Autor:	Khuram Ali Khan, Tasadduq Niaz, Ðilda Pec̆arić, Josip Pec̆arić
Jazyk:	angličtina
Rok vydání:	2018
Předmět:	m-convex function Jensen’s inequality f- and Rényi divergence Montgomery identity Entropy Mathematics QA1-939
Zdroj:	Journal of Inequalities and Applications, Vol 2018, Iss 1, Pp 1-22 (2018)
Druh dokumentu:	article
ISSN:	1029-242X
DOI:	10.1186/s13660-018-1902-9
Popis:	Abstract Jensen’s inequality is important for obtaining inequalities for divergence between probability distribution. By applying a refinement of Jensen’s inequality (Horváth et al. in Math. Inequal. Appl. 14:777–791, 2011) and introducing a new functional based on an f-divergence functional, we obtain some estimates for the new functionals, the f-divergence, and Rényi divergence. Some inequalities for Rényi and Shannon estimates are constructed. The Zipf–Mandelbrot law is used to illustrate the result. In addition, we generalize the refinement of Jensen’s inequality and new inequalities of Rényi Shannon entropies for an m-convex function using the Montgomery identity. It is also given that the maximization of Shannon entropy is a transition from the Zipf–Mandelbrot law to a hybrid Zipf–Mandelbrot law.
Databáze:	Directory of Open Access Journals
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