Entropy (May 2019)

A Direct Link between Rényi–Tsallis Entropy and Hölder’s Inequality—Yet Another Proof of Rényi–Tsallis Entropy Maximization

  • Hisa-Aki Tanaka,
  • Masaki Nakagawa,
  • Yasutada Oohama

DOI
https://doi.org/10.3390/e21060549
Journal volume & issue
Vol. 21, no. 6
p. 549

Abstract

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The well-known Hölder’s inequality has been recently utilized as an essential tool for solving several optimization problems. However, such an essential role of Hölder’s inequality does not seem to have been reported in the context of generalized entropy, including Rényi−Tsallis entropy. Here, we identify a direct link between Rényi−Tsallis entropy and Hölder’s inequality. Specifically, we demonstrate yet another elegant proof of the Rényi−Tsallis entropy maximization problem. Especially for the Tsallis entropy maximization problem, only with the equality condition of Hölder’s inequality is the q-Gaussian distribution uniquely specified and also proved to be optimal.

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