On Zipf–Mandelbrot entropy Adil Khan, Muhammad; Pečarić, Đilda; Pečarić, Josip
Journal of computational and applied mathematics,
01/2019, Volume:
346
Journal Article
Peer reviewed
Open access
Zipf–Mandelbrot entropy has many applications in many applied sciences, for example, in information theory, ecology and economics, etc. The main purpose of this paper is to associate results for ...convex functions with Zipf–Mandelbrot entropy. First of all, we present some inequalities for convex functions and their applications for Csiszar divergence. Then we use two classes of convex functions and apply inequalities for convex functions and establish several results for Zipf–Mandelbrot entropy. Related results for Zipf–Mandelbrot entropies corresponding to different parameters have been proved.
The Jensen’s inequality plays a crucial role to obtain inequalities for divergences between probability distributions. In this paper, we introduce a new functional, based on the
f
-divergence ...functional, and then, we obtain some estimates for the new functional, the
f
-divergence and the Rényi divergence by applying a cyclic refinement of the Jensen’s inequality. Some inequalities for Rényi and Shannon entropies are obtained too. Zipf–Mandelbrot law is used to illustrate the results.
In this paper, four new Green functions are used to generalize Levinson-type inequalities for the class of 3-convex functions. The f-divergence, Renyi entropy, Renyi divergence, Shannon entropy, and ...the Zipf–Mandelbrot law are also used to apply the main results to information theory.
In this work, Levinson type inequalities involving two types of data points are proved using Green functions and the Hermite interpolating polynomial for the class of
n
-convex functions. In seek of ...applications to information theory some estimates for new functionals are obtained, based on
f
-divergence. Moreover, some inequalities involving Shannon entropies are deduced as well.
In this paper, we consider the definition of “useful” Csiszár divergence and “useful” Zipf-Mandelbrot law associated with the real utility distribution to give the results for majorizatioQn ...inequalities by using monotonic sequences. We obtain the equivalent statements between continuous convex functions and Green functions via majorization inequalities, “useful” Csiszár functional and “useful” Zipf-Mandelbrot law. By considering “useful” Csiszár divergence in the integral case, we give the results for integral majorization inequality. Towards the end, some applications are given.
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