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Mathematical inequalities for some divergences

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  • Furuichi, Shigeru
  • Mitroi, Flavia-Corina

Abstract

Some natural phenomena are deviating from standard statistical behavior and their study has increased interest in obtaining new definitions of information measures. But the steps for deriving the best definition of the entropy of a given dynamical system remain unknown. In this paper, we introduce some parametric extended divergences combining Jeffreys divergence and Tsallis entropy defined by generalized logarithmic functions, which lead to new inequalities. In addition, we give lower bounds for one-parameter extended Fermi–Dirac and Bose–Einstein divergences. Finally, we establish some inequalities for the Tsallis entropy, the Tsallis relative entropy and some divergences by the use of the Young’s inequality.

Suggested Citation

  • Furuichi, Shigeru & Mitroi, Flavia-Corina, 2012. "Mathematical inequalities for some divergences," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(1), pages 388-400.
    • Handle: RePEc:eee:phsmap:v:391:y:2012:i:1:p:388-400
    DOI: 10.1016/j.physa.2011.07.052
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    References listed on IDEAS

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    1. Suyari, Hiroki, 2006. "Mathematical structures derived from the q-multinomial coefficient in Tsallis statistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 368(1), pages 63-82.
    2. G. Kaniadakis, 2009. "Maximum entropy principle and power-law tailed distributions," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 70(1), pages 3-13, July.
    3. Suyari, Hiroki & Wada, Tatsuaki, 2008. "Multiplicative duality, q-triplet and (μ,ν,q)-relation derived from the one-to-one correspondence between the (μ,ν)-multinomial coefficient and Tsallis entropy Sq," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(1), pages 71-83.
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    Cited by:

    1. Răzvan-Cornel Sfetcu & Vasile Preda, 2023. "Fractal Divergences of Generalized Jacobi Polynomials," Mathematics, MDPI, vol. 11(16), pages 1-12, August.
    2. Sfetcu, Răzvan-Cornel, 2016. "Tsallis and Rényi divergences of generalized Jacobi polynomials," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 460(C), pages 131-138.
    3. Kluza, Paweł & Niezgoda, Marek, 2016. "Generalizations of Crooks and Lin’s results on Jeffreys–Csiszár and Jensen–Csiszár f-divergences," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 463(C), pages 383-393.
    4. Qiu, Guoxin & Jia, Kai, 2018. "The residual extropy of order statistics," Statistics & Probability Letters, Elsevier, vol. 133(C), pages 15-22.
    5. Popescu, P.G. & Preda, V. & Sluşanschi, E.I., 2014. "Bounds for Jeffreys–Tsallis and Jensen–Shannon–Tsallis divergences," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 413(C), pages 280-283.
    6. Kluza, Paweł A., 2020. "On Jensen–Rényi and Jeffreys–Rényi type f-divergences induced by convex functions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 548(C).
    7. Guoxin Qiu & Kai Jia, 2018. "Extropy estimators with applications in testing uniformity," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 30(1), pages 182-196, January.

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