Estimation of different entropies via Lidstone polynomial using Jensen-type functionals
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Arabian Journal of Mathematics
Khuram Ali Khan · Tasadduq Niaz Josip Peˇcari´c
· -Dilda Peˇcari´c ·
Estimation of different entropies via Lidstone polynomial using Jensen-type functionals
Received: 5 November 2018 / Accepted: 1 February 2020 © The Author(s) 2020
Abstract In this work, some new functional of Jensen-type inequalities are constructed using Shannon entropy, f -divergence, and Rényi divergence, and some estimates are obtained for these new functionals. Also using the Zipf–Mandelbrot law and hybrid Zipf–Mandelbrot law, we investigate some bounds for these new functionals. Furthermore, we generalize these new functionals for m-convex function using Lidstone polynomial. Mathematics Subject Classification
94Axx · 39-XX · 41A58
1 Introduction and preliminary results The most commonly used words, population ranks of cities in various countries, corporation sizes, income rankings can be described in terms of Zipf’s law. The f -divergence measures the difference between two probability distributions by making an average value, which is weighted by a specified function. There are other probability distributions like Csiszar f -divergence [10,11], a special case of which is Kullback–Leiblerdivergence which is used to find the appropriate distance between the probability distributions (see [18, 19]). The notion of distance is stronger than divergence, because it gives the properties of symmetry and triangle inequalities. Probability theory has application in many fields and the divergence between probability distributions has many applications in these fields. The research of fourth author was supported by the Ministry of Education and Science of the Russian Federation (the Agreement number No. 02.a03.21.0008). K. A. Khan · T. Niaz (B) Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan E-mail: [email protected] K. A. Khan E-mail: [email protected] T. Niaz Department of Mathematics, The University of Lahore, Sargodha-Campus, Sargodha 40100, Pakistan -Dilda Peˇcari´c Catholic University of Croatia, Ilica, 242 Zagreb, Croatia E-mail: [email protected] Josip Peˇcari´c RUDN University, Miklukho-Maklaya str. 6, 117198 Moscow, Russia E-mail: [email protected]
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Arab. J. Math.
Many natural phenomena such as distributions of wealth and income in a society, Facebook likes, football goals, and city sizes follow power-law distributions (Zipf’s Law). Auerbach [2] was the first to explore the idea that the distribution of city size can be well approximated with the help of Pareto distribution (power-law distribution). This idea was well refined by many researchers, but Zipf [27] worked significantly in this field. The distribution of city sizes is investigated by many scholars of the urban economics, like Rosen and Resnick [24], Black and Henderson [3], Ioannides and Overman [17], Soo [25], Anderson and Ge [1], and Bosker et al. [4]. Zipf’s law states that: “The rank of cities with a certain number of inhabitants varies proportional to the city sizes with some negative exponent,
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