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

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 som...

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Bibliographic Details
Published in:Journal of inequalities and applications 2018, Vol.2018 (1), p.1-22, Article 318
Main Authors: Khan, Khuram Ali, Niaz, Tasadduq, Pec̆arić, Ðilda, Pec̆arić, Josip
Format: Article
Language:English
Subjects:
Analysis
Applications of Mathematics
Divergence
Entropy
Entropy (Information theory)
f- and Rényi divergence
Inequalities
Inequality
Jensen’s inequality
m-convex function
Mathematics
Mathematics and Statistics
Montgomery identity
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Summary: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.
ISSN:1029-242X
1025-5834
1029-242X
DOI:10.1186/s13660-018-1902-9