Refinement of Jensen’s inequality and estimation of f- and Rényi divergence via Montgomery identity

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.

Generalization of the Levinson inequality with applications to information theory

“…A variety of sorts of divergences exist, for instance the fdifference (particularly, the Kullback-Leibler divergence, the Hellinger distance and the total variation distance), the Rényi divergence, the Jensen-Shannon divergence, and so forth (see [13,21]). There are a lot of papers managing inequalities and entropies, see, e.g., [8,10,20] and the references therein. The Jensen inequality assumes a crucial role in a part of these inequalities.…”

RETRACTED ARTICLE: Generalization of the Levinson inequality with applications to information theory

“…First we give an identity involving Jensen's difference of two different data points. Then we give an equivalent form of identity by using the Green's function defined by (10) and (12).…”

“…The presented work is organized as follows: In Sect. 2, Levinson's inequality for 3-convex function is generalized by using two Green's functions defined by (10) and (12). In Sect.…”

“…where G k (•, s) are defined by (10) and ( (9). Now, by means of some simple calculations, we can write…”

Estimation of Different Entropies via Taylor One Point and Taylor Two Points Interpolations Using Jensen Type Functionals

“…In [23] T. Niaz et al generalized the refinement of Jensen's inequality for m-convex function using Abel-Gontscharoff green function and Fink's identity. In [18] Since many years Jensen's inequality has of great interest. The researchers have given the refinement of Jensen's inequality by defining some new functions (see [12,13] ).…”