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

Graduate School of Informatics and Engineering, The University of Electro-Communications, Tokyo 182-8585, Japan
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Entropy 2019, 21(6), 549; https://doi.org/10.3390/e21060549
Received: 30 April 2019 / Revised: 27 May 2019 / Accepted: 27 May 2019 / Published: 30 May 2019
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Abstract

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. View Full-Text
Keywords: Rényi–Tsallis entropy; generalized entropy; optimization; Hölder’s inequality Rényi–Tsallis entropy; generalized entropy; optimization; Hölder’s inequality
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Tanaka, H.-A.; Nakagawa, M.; Oohama, Y. A Direct Link between Rényi–Tsallis Entropy and Hölder’s Inequality—Yet Another Proof of Rényi–Tsallis Entropy Maximization. Entropy 2019, 21, 549.

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