On the equality problem of generalized Bajraktarević means

Grünwald, Richárd and Páles, Zsolt (2020) On the equality problem of generalized Bajraktarević means. AEQUATIONES MATHEMATICAE, 94 (4). pp. 651-677. ISSN 0001-9054

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Abstract

The purpose of this paper is to investigate the equality problem of generalized Bajraktarevic means, i.e., to solve the functional equation f((-1))(p(1)(x(1))f(x(1))+ ... + p(n)(x(n))f(x(n))/p(1)(x(1))+ ... +p(n)(x(n))) = g((-1))(q(1)(x(1))g(x(1))+ ... +q(n)(x(n))g(x(n))/q(1)(x(1))+ ... +q(n)(x(n))), (*) which holds for all (x(1), ... , x(n)) is an element of I-n, where n >= 2, I is a nonempty open real interval, the unknown functions f, g : I -> R are strictly monotone, f((-1)) and g((-1)) denote their generalized left inverses, respectively, and p = (p(1), ... , p(n)) : I -> R-+(n) and q = (q(1), ... , q(n)) : I -> R-+(n) are also unknown functions. This equality problem in the symmetric two-variable (i.e., when n = 2) case was already investigated and solved under sixth-order regularity assumptions by Losonczi (Aequationes Math 58(3):223-241, 1999). In the nonsymmetric two-variable case, assuming the three times differentiability of f, g and the existence of i is an element of {1, 2} such that either p(i) is twice continuously differentiable and p(3-i) is continuous on I, or p(i) is twice differentiable and p(3-i) is once differentiable on I, we prove that (*) holds if and only if there exist four constants a, b, c, d is an element of R with ad not equal bc such that cf + d > 0, g = af + b/cf + d, and q(l) = (cf + d)p(l) (l is an element of {1, ... , n}). In the case n >= 3, we obtain the same conclusion with weaker regularity assumptions. Namely, we suppose that f and g are three times differentiable, p is continuous and there exist i, j, k is an element of {1, ... , n} with i not equal j not equal k not equal i such that p(i), p(j), p(k) are differentiable.

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