### 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 (x_{1}, ⋯ , x_{n}) ∈ 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=(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 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∈ 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 p_{i}, p_{j}, p_{k} are differentiable.

Original language | English |
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Journal | Aequationes Mathematicae |

DOIs | |

Publication status | Published - 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