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Estimations of f- and Rényi Divergences by Using a Cyclic Refinement of the Jensen’s Inequality

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Abstract

The Jensen’s inequality plays a crucial role to obtain inequalities for divergences between probability distributions. In this paper, we introduce a new functional, based on the f-divergence functional, and then, we obtain some estimates for the new functional, the f-divergence and the Rényi divergence by applying a cyclic refinement of the Jensen’s inequality. Some inequalities for Rényi and Shannon entropies are obtained too. Zipf–Mandelbrot law is used to illustrate the results.

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Acknowledgements

The research of the first author has been supported by Hungarian National Foundations for Scientific Research Grant No. K120186.

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Authors and Affiliations

  1. Department of Mathematics, University of Pannonia, Egyetem u. 10, Veszprém, 8200, Hungary

    László Horváth

  2. Catholic University of Croatia, Ilica 242, 10000, Zagreb, Croatia

    Đilda Pečarić

  3. Faculty of Textile Technology, University of Zagreb, Pierottijeva 6, 10000, Zagreb, Croatia

    Josip Pečarić

Authors
  1. László Horváth
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  2. Đilda Pečarić
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  3. Josip Pečarić
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Corresponding author

Correspondence to László Horváth.

Additional information

Communicated by Fuad Kittaneh.

The research of the first author has been supported by Hungarian National Foundations for Scientific Research Grant No. K120186.

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Horváth, L., Pečarić, Đ. & Pečarić, J. Estimations of f- and Rényi Divergences by Using a Cyclic Refinement of the Jensen’s Inequality. Bull. Malays. Math. Sci. Soc. 42, 933–946 (2019). https://doi.org/10.1007/s40840-017-0526-4

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  • DOI: https://doi.org/10.1007/s40840-017-0526-4

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