A generalization of HH $f$-divergence

Highlights

• The fractional HH $f$-divergence is introduced.

• We combine three well-known fractional integrals with HH $f$-divergence.

• Our results generalize the corresponding ones in the literature.

Abstract

The discrimination between two probability distributions is an impotent problem. In 1991, Lin [IEEE Transactions on Information Theory, 37(1) 1991] introduced a novel class of information-theoretic divergence measures based on the Shannon entropy. As a generalization of Lin’s divergence, a new divergence, called Hermite–Hadamard (HH) $f$-divergence, based on Lin’s method of constructing the divergence was introduced by Shioya and Da-te in 1995. In this paper, we expand the applicability of HH $f$-divergence by combining the properties of fractional calculus with HH $f$-divergence, and then introduce the concept of some fractional HH $f$-divergences which are generalizations of the HH $f$-divergence. Then, some inequalities related to fractional HH $f$-divergence are proposed.