The concept of superquadratic functions in several variables, as a generalization of the same concept in one variable is introduced. Analogous results to results obtained for convex functions in one and several variables are presented. These include refinements of Jensen's inequality and its counterpart, and of SlaterPečarić's inequality.

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In this paper we give Bessel-and Grüss-type inequalities in an inner product module over a proper H * -algebra or a C * -algebra.

Abstract. Refinements of Jensen-Steffensen's inequality, Slater-Pečarić's inequality and majorization theorems for superquadratic functions are presented.Mathematics subject classification (2000): 26A51, 26D15, 26D20.

Abstract. Using some characterizations of superquadratic functions we obtain some results on two mappings H and F connected with Hermite-Hadamard inequality. Our results generalize corresponding results for convex functions. Especially, when the superquadratic function is convex at the same time, then our results represent also the refinements of those results. (2000): 26D15. Mathematics subject classification