On the equality problem of generalized Bajraktarević means

Richárd Grünwald, Z. Páles

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Abstract

The purpose of this paper is to investigate the equality problem of generalized Bajraktarević means, i.e., to solve the functional equation f(-1)(p1(x1)f(x1)+⋯+pn(xn)f(xn)p1(x1)+⋯+pn(xn))=g(-1)(q1(x1)g(x1)+⋯+qn(xn)g(xn)q1(x1)+⋯+qn(xn)),which holds for all (x1, ⋯ , xn) ∈ In, 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=(p1,⋯,pn):I→R+n and q=(q1,⋯,qn):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∈ { 1 , 2 } such that either pi is twice continuously differentiable and p3 - i is continuous on I, or pi is twice differentiable and p3 - i is once differentiable on I, we prove that (*) holds if and only if there exist four constants a, b, c, d∈ R with ad≠ bc such that cf+d>0,g=af+bcf+d,andqℓ=(cf+d)pℓ(ℓ∈{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∈ { 1 , ⋯ , n} with i≠ j≠ k≠ i such that pi, pj, pk are differentiable.

Original languageEnglish
JournalAequationes Mathematicae
DOIs
Publication statusPublished - Jan 1 2019

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Keywords

  • Bajraktarević mean
  • Equality of means
  • Generalized inverse
  • Quasi-arithmetic mean

ASJC Scopus subject areas

  • Mathematics(all)
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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