Abstract
Zipf–Mandelbrot and Shannon entropies are some basic and useful tools to quantify information about certain phenomena in various fields of science and technology, for example statistics, ecology, biology, and information theory. In this paper, we obtain some new bounds for generalized Shannon and Zipf–Mandelbrot entropies by using some specific refinements of Jensen’s inequality. Then, as a consequence of these bounds, we deduce some new bounds for Zipf–Mandelbrot and Shannon entropies. Finally, we demonstrate the sharpness of the proposed bounds through numerical experiments.
Highlights
Information theory is a useful mathematical tool, not limited to communication, but more technically as an important part of probability theory
Theorem 8 provides some new bounds for the generalized Shannon entropy, which are obtained from Theorem 1
Theorem 9 proposes some new bounds for the generalized Shannon entropy, which are the applications of Theorem 2
Summary
Information theory is a useful mathematical tool, not limited to communication, but more technically as an important part of probability theory. The Shannon entropy was the beginning of information theory, and different entropic functionals have been developed since . These functionals can be viewed as the generalizations of entropic measures and are appropriate in various situations. Zipf was the first to study the distribution of words in large texts by ranking the words in decreasing order of frequency. The Zipf–Mandelbrot law gives a best fit as compared to Zipf’s law. We cite some refinements of Jensen’s inequality We employ these refinements to obtain some new bounds for generalized Zipf–Mandelbrot and generalized Shannon entropies. The following theorem provides an upper bound for Jensen’s gap:
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