A Pexider equation containing the aritmetic mean

In this paper we determine the solutions ( φ , f 1 , f 2 ) of the Pexider functional equation φ ( x + y 2 ) ( f 1 ( x ) - f 2 ( y ) ) = 0 , ( x , y ) ∈ I 1 × I 2 , where I 1 and I 2 are nonempty open subintervals. Special cases of the above equation regularly arise in problems with two-variable mean...

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Bibliographic Details
Published inAequationes mathematicae Vol. 98; no. 2; pp. 579 - 589
Main Author Kiss, Tibor
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.04.2024
Springer Nature B.V
Subjects
Analysis
Combinatorics
Functional equations
Mathematics
Mathematics and Statistics
Regularity
Secondary 26E60
Arithmetic mean
Primary 39B22
Functional equation
Pexider equation
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Summary:In this paper we determine the solutions ( φ , f 1 , f 2 ) of the Pexider functional equation φ ( x + y 2 ) ( f 1 ( x ) - f 2 ( y ) ) = 0 , ( x , y ) ∈ I 1 × I 2 , where I 1 and I 2 are nonempty open subintervals. Special cases of the above equation regularly arise in problems with two-variable means. We show that, under a rather weak regularity condition, the coordinate-functions of a typical solution of the equation are constant over several subintervals of their domain. The regularity condition in question will be that the set of zeros of φ is closed. We also discuss particular solutions where this condition is not met.
ISSN:0001-9054
1420-8903
DOI:10.1007/s00010-023-00966-x