Uniform Treatment of Jensen’s Inequality by Montgomery Identity

Abstract

We generalize Jensen’s integral inequality for real Stieltjes measure by using Montgomery identity under the effect of $\mathfrak{n}-$ convex functions; also, we give different versions of Jensen’s discrete inequality along with its converses for real weights. As an application, we give generalized variants of Hermite–Hadamard inequality. Montgomery identity has a great importance as many inequalities can be obtained from Montgomery identity in $q-$ calculus and fractional integrals. Also, we give applications in information theory for our obtained results, especially for Zipf and Hybrid Zipf–Mandelbrot entropies.