Abstract
Jensen’s inequality is important for obtaining inequalities for divergence between probability distribution. By applying a refinement of Jensen’s inequality (Horváth et al. in Math. Inequal. Appl. 14:777–791, 2011) and introducing a new functional based on an f-divergence functional, we obtain some estimates for the new functionals, the f-divergence, and Rényi divergence. Some inequalities for Rényi and Shannon estimates are constructed. The Zipf–Mandelbrot law is used to illustrate the result. In addition, we generalize the refinement of Jensen’s inequality and new inequalities of Rényi Shannon entropies for an m-convex function using the Montgomery identity. It is also given that the maximization of Shannon entropy is a transition from the Zipf–Mandelbrot law to a hybrid Zipf–Mandelbrot law.
Highlights
Introduction and preliminary resultsThe most commonly used words, the largest cities of countries, income of a billionaire can be described in terms of Zipf ’s law
The f -divergence means the distance between two probability distributions by making an average value, which is weighted by a specified function
The notion of distance is stronger than that of divergence because it gives the properties of symmetry and triangle inequalities
Summary
Introduction and preliminary resultsThe most commonly used words, the largest cities of countries, income of a billionaire can be described in terms of Zipf ’s law. The researchers have given the refinement of Jensen’s inequality by defining some new functions (see [16, 18]). Theorem 1.1 Assume (H1), and let f : I → R be a convex function where I ⊂ R is an interval.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
