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


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Article Info

DATE

2018-11-19

AUTHORS

Khuram Ali Khan, Tasadduq Niaz, Ðilda Pec̆arić, Josip Pec̆arić

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. More... »

PAGES

318

References to SciGraph publications

  • 2000. Shannon’s and Related Inequalities in Information Theory in SURVEY ON CLASSICAL INEQUALITIES
  • 2018-02-08. Zipf–Mandelbrot law, f-divergences and the Jensen-type interpolating inequalities in JOURNAL OF INEQUALITIES AND APPLICATIONS
  • 2017-07-28. Estimations of f- and Rényi Divergences by Using a Cyclic Refinement of the Jensen’s Inequality in BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY
  • 1993. Classical and New Inequalities in Analysis in NONE
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1186/s13660-018-1902-9

    DOI

    http://dx.doi.org/10.1186/s13660-018-1902-9

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1110037209

    PUBMED

    https://www.ncbi.nlm.nih.gov/pubmed/30839843


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