Refinement of Jensen’s inequality and estimation of f- and Rényi divergence via Montgomery identity
| Autor: | Tasadduq Niaz, Khuram Ali Khan, Ðilda Pečarić, Josip Pečarić |
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| Rok vydání: | 2018 |
| Předmět: |
Discrete mathematics
lcsh:Mathematics Research Entropy Applied Mathematics 010102 general mathematics Montgomery identity Maximization lcsh:QA1-939 01 natural sciences Jensen’s inequality f- and Rényi divergence 010101 applied mathematics m-convex function Discrete Mathematics and Combinatorics Entropy (information theory) Probability distribution 0101 mathematics Jensen's inequality Analysis Mathematics |
| Zdroj: | Journal of Inequalities and Applications Journal of Inequalities and Applications, Vol 2018, Iss 1, Pp 1-22 (2018) |
| ISSN: | 1029-242X |
| DOI: | 10.1186/s13660-018-1902-9 |
| Popis: | 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: | OpenAIRE |
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