Extension Theorems for the Matkowski-Suto Problem

Abstract

Abstract. In this paper the functional equation is studied, where <p, ip are continuous strictly monotonic functions on the open real interval I. Under analicity assumptions (*) was first solved by O. Suto in 1914. Since then several efforts have been made to weaken the regularity assumptions on the unknown functions. In this paper we show an extension type theorem for (*) and obtain solutions of (*) assuming weak regularity properties of ip and i/> 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.

Problems in the regularity theory of functional equations

“…In another paper, Daróczy and Maksa [DM99] have solved (11) assuming that the two unknown functions are convex or concave. Finally, Daróczy, Maksa, and Páles [DMP00] have found an extension theorem, from which one can obtain the same conclusion under the assumption that one of the unknown functions is continuously differentiable on a proper subinterval of I. It is a natural conjecture that the solutions of (11) without any additional regularity conditions are the same as in the cases described above.…”

On linear combinations of weighted quasi-arithmetic means

“…Let f 1 (x) = χ p (x), f 2 (x) = χ −p (x) (x ∈ I) for some 0 = p ∈ R. This case was studied by Daróczy, Maksa and Páles [4].…”

The invariance of the arithmetic mean with respect to generalized quasi-arithmetic means

“…These regularity assumptions were weakened step-by-step by Daróczy, Maksa, and Páles in the papers [6,7], and finally in 2002 the following result was proved [8]:…”