A Matkowski--Sutô type equation

Abstract

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.

On the equality of generalized quasi-arithmetic means

“…Due to the implicit function theorem, f is continuously differentiable on [x, z]. Differentiating (10) with respect to the variable t, it follows that…”

“…A recently rediscovered and blossoming subject is the investigation of the so-called invariance equation and the Gauss-iteration related to quasi-arithmetic means: Gauss [33], B lasińska-Lesk-G lazowska-Matkowski [7], Burai [9], [10], Daróczy [11]- [15], Daróczy-Hajdu [16], Daróczy-Hajdu-Ng [17], Daróczy-Lajkó-Lovas-Maksa-Páles [18], Daróczy-Maksa [19], Daróczy-Maksa-Páles [20], [22], Daróczy-Ng [23], Daróczy-Páles [25], [27], [26], [28], [30], [24], [29], Domsta-Matkowski [32], G lazowska-Jarczyk-Matkowski [34], Hajdu [35], Haruki-Rassias [37], Jarczyk-Matkowski [40], Jarczyk [38], Matkowski [47], [48], [50],…”

On the invariance of the arithmetic mean with respect to generalized Bajraktarević means

“…Finally Daróczy and Páles solved this problem without any unnecessary differentiability assumptions in [7]. Jarczyk and Matkowski in [10] and Burai in [6] has studied the invariance equation in case of three weighted arithmetic means. Jarczyk in [9] gave a solution to this problem with no additional regularity assumptions.…”

On a functional equation with a symmetric component

“…(iii) µ = r, (p, q, r) ∈ (0, 1) 3 J. Jarczyk and J. Matkowski in [8], and J. Jarczyk [7], P. Burai [1].…”