Abstract. We are going to prove an extension theorem on a functional equation for special means studied by Domsta and Matkowski.

No abstract

In this paper we show that bisymmetry, which is an algebraic property, has a regularity improving feature. More precisely, we prove that every bisymmetric, partially strictly monotonic, reflexive and symmetric function F : I 2 → I is continuous. As a consequence, we obtain a finer characterization of quasiarithmetic means than the classical results of Aczél [1], Kolmogoroff [18], Nagumo [20] and de Finetti [11].

The main goal of this paper is to prove some new results and extend some earlier ones about functions, which possess the so called local-global minimum property. In the last section, we show an application of these in the theory of calculus of variations.

In the present paper we deal with the following equation:where ϕ and ψ are strictly monotone and continuous functions on the same interval. We give the continuously differentiable solutions. Definition 1. Let I ⊂ R be a nonempty open interval and let CM(I) denote the class of all continuous, strictly monotone functions defined on I. Definition 2. A continuous function M : I 2 → I is called a mean on I if min{x, y} ≤ M (x, y) ≤ max{x, y} for all x, y ∈ I.