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In 1985 Godunova and Levin have considered the following class of functions. A function f : I → R is said to belong to the class Q (I) if it is nonnegative and for all x,y ∈ I and t ∈ (0,1) , satisfies the inequality: f ((1 − t) x + ty) f (x) 1 − t + f (y) t Here I is an interval of R. It is known that all nonnegative quasiconvex functions belong to this class and this class of functions coincides with the class of Schur functions S (I) , that is, with the class of nonnegative functions that satisfy the inequality ∑ f (x) (x − y) (x − z) 0 f o re v e r y x,y,z ∈ I The aim of this paper is to survey some important properties of functions belonging to these classes of functions and to prove some new results concerning properties of functions from them.

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In this paper a refinement of a particular case of an inequality of Radon, referred in the literature as Bergström's inequality, is presented. Later on, this result is used to obtain refinements of some known inequalities among them the classical inequality of Cauchy. Finally, two elementary numerical inequalities are also given. (2000): 26D15. Mathematics subject classification