On a functional equation related to two-variable weighted quasi-arithmetic means

Abstract

In this paper, we are going to describe the solutions of the functional equation $$\phi \left(\frac{x+y}{2}\right)(f(x)+f(y))=\phi (x)f(x)+\phi (y)f(y)$$ concerning the unknown functions $\phi$ and $f$ defined on an open interval. In our main result only the continuity of the function $\phi$ and a regularity property of the set of zeroes of $f$ are assumed. As application, we determine the solutions of the functional equation $$G(g(u)-g(v))=H(h(u)+h(v))+F(u)+F(v)$$ under monotonicity and differentiability conditions on the unknown functions $F,G,H,g,h$.

References in corpus (3)

• Characterization of generalized quasi-arithmetic means
• On the commutation of generalized means on probability spaces
• Explicit solutions of the invariance equation for means

Cited by in corpus (4)

• On the equality of Bajraktarević means to quasi-arithmetic means
• On the equality of two-variable general functional means
• Characterization of the equality of Cauchy means to quasiarithmetic means
• Permuting triderivations and permuting trihomomorphisms in complex Banach algebras