Generalization of cyclic refinements of Jensen’s inequality by Fink’s identity

Abstract

We generalize cyclic refinements of Jensen's inequality from a convex function to a higher-order convex function by means of Lagrange-Green's function and Fink's identity. We formulate the monotonicity of the linear functionals obtained from these identities utilizing the theory of inequalities for n-convex functions at a point. New Grüss-and Ostrowski-type bounds are found for identities associated with the obtained inequalities. Finally, we investigate the properties of linear functionals regarding exponential convexity and mean value theorems.MSC: Primary 26A51; 26D07; 26D15; 26D20; 26D99