Naslov
Zipf-Mandelbrot law, Properties and its generalizations

Autori
Jakšetić, Julije ; Pečarić, Đilda ; Pečarić, Josip

Vrsta, podvrsta i kategorija rada
Poglavlja u knjigama, znanstveni

Knjiga
Inequalities and Zipf-Mandelbrot law. Selected topics in information theory, Monographs in inequalities 15

Urednik/ci
Đilda Pečarić, Josip Pečarić

Izdavač
Element

Grad
Zagreb

Godina
2019

Raspon stranica
1-28

ISBN
978-953197-670-1

Ključne riječi
Zipf-Mandelbrot law, inequalities, Shannon entropy, Chebyshev’s and Lyapunov’s inequality

Sažetak
Despite a wide spread applications of Zipf-Mandelbrot law, there is quite small amount of results concerning analytical properties on distribution law. On the first stage, we examine some monotonicity properties of the law, we derive the whole variety of its lower and upper estimations. We then further refine our results using some well-known inequalities such as H¨older and Lyapunov inequality. On the second stage we consider the case when total mass of Zipf-Mandelbrot law is spread all over positive integer, and then we come to Hurwitz −function. As we show, it is very natural first to examine properties of Hurwitz −function to derive properties of Zipf-Mandelbrot law. Using some well-known inequalities such as Chebyshev’s and Lyapunov’s inequality we are able to deduce a whole variety of theoretical characterizations that include, among others, log-convexity, log-subadditivity, exponential convexity. On the third stage, we generalize Zipf-Mandelbrot law using maximization of Shannon entropy, as we get hybrid Zipf-Mandelbrot law. It is interesting that examination of its densities provides some new insights of Lerch’s transcendent.

Izvorni jezik
Engleski

Znanstvena područja
Matematika, Interdisciplinarne prirodne znanosti, Informacijske i komunikacijske znanosti, Interdisciplinarne društvene znanosti

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