Generalizations of Cyclic Refinements of Jensen’s Inequalities

Abstract

In recent years, the concept of convex functions has been generalized extensively. Applications of convex functions are widely seen in many areas of modern analysis. Convex functions also have significant relation with the theory of inequalities and many useful inequalities are the result of the applications of convex functions. The Jensen's inequality has tremendous implications in many fields of modern analysis. It helps computing useful upper bounds for several entropic measures used in information theory. We consider discrete and continuous cyclic refinements of Jensen's inequality and extend them from convex to higher order convex function by using new Green functions introduced by us and employing different interpolating polynomials and identities. We formulate monotonicity of the linear functionals for nconvex functions at a point. We calculate some new Grüss and Ostrowski type bounds. As an application of our obtained results we give new bounds for Shannon, Relative and Zipf-Mandelbrot entropies.