Abstract. In this paper, the functional equationis considered, where 0 < p < 1 is a fixed parameter and f : I → R is an unknown function. The equivalence of this and Jensen's functional equation is completely characterized in terms of the algebraic properties of the parameter p. As an application, solutions of certain functional equations involving four weighted arithmetic means are also determined.

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In this note we solve the equality problem of weighted quasi-arithmetic means of two variables in the particular case when one of the weight functions is constant. As an application, we determine all those weighted arithmetic means that are also quasi-arithmetic. No differentiability conditions on the unknown functions are assumed.

Abstract. In this paper the functional equation is studied, where in a small subinterval of I. History of the ProblemIn 1914 O. Suto published a paper of two parts in the Tohoku Mathematical Journal [11], in which he first examined the functional equationThe author writes the following: "The analytic functions tp,ip and their inverses are supposed to be one-valued, as ever we do;is a mean of x and y in certain sense." In 1998 the functional equation (1.1) was re-discovered by J. Matkowski [8], who formulated the problem more exactly in the following way.1991 Mathematics Subject Classification: 39B22, 39B12, 26A18.

In this paper we consider higher-order Wright-convex functions and prove that they are representable as the sum of a continuous higher-order convex function and a polynomial function.