Abstract
We present some fundamental results and definitions regarding Jensen’s inequality with the aim of obtaining new generalizations of cyclic refinements of Jensen’s inequality from convex to higher order convex functions using Taylor’s formula. We discuss the monotonicity of functionals for n-convex functions at a point. Applications of our work include new bounds for some important inequalities used in information theory.
Highlights
1 Introduction The convexity of functions has been frequently used in various fields of pure and applied mathematics, for instance in function theory, mathematical analysis, functional analysis, probability theory, optimization theory, operational research, information theory
The simple generalization to a convex function extensively widens our scope for analysis
The Jensen inequality for convex functions plays a pivotal role in the theory of inequalities because of the fact that various other inequalities, for instance the Holder and Minkowski inequalities and the arithmetic mean–geometric mean inequality can be obtained as particular cases
Summary
The convexity of functions has been frequently used in various fields of pure and applied mathematics, for instance in function theory, mathematical analysis, functional analysis, probability theory, optimization theory, operational research, information theory. Remark 1 Under the assumptions of Theorem 1 and Theorem 2 for ψ to be a convex function, we have m The following theorem gives a key criterion to test the n-convexity of a function ψ
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