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.
Highlights
Tsallis entropy [1,2] has been recently utilized as a versatile framework for expanding the realm of Shannon–Boltzmann entropy for nonlinear processes, in particular, those that exhibit power–law behavior
We find a direct link between Rényi–Tsallis entropy and Hölder’s inequality that leads to yet another elegant proof of Rényi–Tsallis entropy maximization
We focus on the univariate probability density functions (PDFs) on R, and we consider f ( x ) and g( x ) defined on
Summary
Tsallis entropy [1,2] has been recently utilized as a versatile framework for expanding the realm of Shannon–Boltzmann entropy for nonlinear processes, in particular, those that exhibit power–law behavior. We note that Hölder’s inequality has been recently utilized as an essential tool for optimization in Campbell [6], Bercher [8], and Bunte and Lapidoth [9]; on source coding, in Bercher [33,34]; on generalized Cramér–Rao inequalities; and in Tanaka [35,36] on a physical limit of injection locking Such an essential role of Hölder’s inequality does not seem to be reported in the context of generalized entropy, including Rényi entropy (cf [37]), except for the use as a means for proving nonnegativity of a generalized relative entropy, as mentioned above. One Appendix at the end provides further supplementary information
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