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The intention of this article is to investigate the most important inequalities of m -convex functions without using their derivatives. The article also provides a brief survey of general properties of m -convex functions.

The article provides refinements and generalizations of the Hermite-Hadamard inequality for convex functions on the bounded closed interval of real numbers. Improvements are related to the discrete and integral part of the inequality. MSC: 26A51; 26D15

We give an extension of Jensen's inequality for -tuples of self-adjoint operators, unital -tuples of positive linear mappings, and real-valued continuous convex functions with conditions on the operators' bounds. We also study operator quasiarithmetic means under the same conditions.