Abstract. Using the Ohlin lemma on convex stochastic ordering we prove inequalities of the Hermite-Hadamard type. Namely, we determine all numbers a, α, β ∈ [0, 1] such that for all convex functions f the inequalityis satisfied and all a, b, c, α ∈ (0, 1) with a + b + c = 1 for which we haveMathematics Subject Classification. Primary 26A51; Secondary 26D10, 39B62.

Abstract. We observe that the Hermite-Hadamard inequality written in

We deal with the functional equationmotivated by quadrature rules of approximate integration. In previous results the solutions of this equation were found only in some particular cases. For example the coefficients λi were supposed to be rational or the equation in question was solved only for n = 2. In the current paper we do not assume any particular form of coefficients occurring in this equation and we allow n to be any positive integer. Moreover, we obtain a solution of our equation without any regularity assumptions concerning the functions f and F .

Abstract. A function s(x, y) = inf λ∈R x+λy x strictly connected with Birkhoff-James orthogonality is considered. This function may be used in functional equations theory, to provide unconditional equations in place of orthogonal equations in the sense of Birkhoff-James. Moreover, we deal with another generalization of the sine function which, in particular leads to a characterization of inner product spaces.