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Generalizations and refinements of the Jensen-Mercer inequality

Our starting point is the following variant of Jensen's inequality (A. McD. Mercer, Journal of Inequalities in Pure and Appllied Mathematics, 2003.): f(a+b-(1/(W_{;n};))∑ _{;i=1};ⁿ w_{;i};x_{;i};)≤ f(a)+f(b)-(1/(W_{;n};))∑ _{;i=1};ⁿ w_{;i};f(x_{;i};), for a convex function f:[a, b]→ ℝ , real numbers x₁ , … , x_{;n};∈ [a, b] and positive real numbers w₁ , … , w_{;n};, where W_{;n};=∑ _{;i=1};ⁿ w_{;i};. We call it Jensen-Mercer inequality and we present its generalizations and refinements in various spaces with adequate orders, and for several types real valued functions such as convex functions, P-convex functions, functions with nondecreasing increments, superquadratic functions and operator convex functions.

Jensen-Mercer inequality; convex functions; P-convex functions; functions with nondecreasing increments; superquadratic functions; operator convex functions

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