On the Equality Problem of Conjugate Means

Autor: Zoltán Daróczy, Judita Dascăl
Rok vydání: 2010
Předmět:
Zdroj: Results in Mathematics. 58:69-79
ISSN: 1420-9012
1422-6383
DOI: 10.1007/s00025-010-0042-4
Popis: Let $${I\subset\mathbb{R}}$$ be a nonvoid open interval and let L : I 2→ I be a fixed strict mean. A function M : I 2→ I is said to be an L-conjugate mean on I if there exist $${p,q\in\,]0,1]}$$ and $${\varphi\in CM(I)}$$ such that $$M(x,y):=\varphi^{-1}(p\varphi(x)+q\varphi(y)+(1-p-q) \varphi(L(x,y)))=:L_\varphi^{(p,q)}(x,y),$$ for all $${x,y\in I}$$ . Here L(x, y) : = A χ(x, y) $${(x,y\in I)}$$ is a fixed quasi-arithmetic mean with the fixed generating function $${\chi\in CM(I)}$$ . We examine the following question: which L-conjugate means are weighted quasi-arithmetic means with weight $${r\in\, ]0,1[}$$ at the same time? This question is a functional equation problem: Characterize the functions $${\varphi,\psi\in CM(I)}$$ and the parameters $${p,q\in\,]0,1]}$$ , $${r\in\,]0,1[}$$ for which the equation $$L_\varphi^{(p,q)}(x,y)=L_\psi^{(r,1-r)}(x,y)$$ holds for all $${x,y\in I}$$ .
Databáze: OpenAIRE
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