On the invariance of the arithmetic mean with respect to generalized Bajraktarević means

Abstract

The purpose of this paper is to investigate the following invariance equation involving two 2-variable generalized Bajraktarević means, i.e., we aim to solve the functional equation

$$f^{-1}\Bigl(\frac{p_1(x)f(x) +p_2(y)f(y)}{p_1(x)+p_2(y)}\Bigr)+g^{-1}\Bigl(\frac{q_1(x)g(x) +q_2(y)g(y)}{q_1(x)+q_2(y)}\Bigr)=x + y \ \ (x,y\in I),$$

where I is a nonempty open real interval and $f,g \colon I \to\mathbb{R}$ are continuous, strictly monotone and $p_1,p_2,q_1,q_2 \colon I \to\mathbb{R}_+$ are unknown functions. The main result of the paper shows that, assuming four times continuous differentiability of f, g, twice continuous differentiability of p₁ and p₂ and assuming that p₁ differs from p₂ on a dense subset of I, a necessary and sufficient condition for the equality above is that the unknown functions are of the form

$$f=\frac{u}{v}, \quad g=\frac{w}{z}, \quad \mbox{and} \quad p_1q_1=p_2q_2=vz,$$

where $u,v,w,z \colon I \to\mathbb{R}$ are arbitrary solutions of the second-order linear differential equation $F''=\gamma F (\gamma\in\mathbb{R}$ is arbitrarily fixed) such that v > 0 and z > 0 holds on I and $\{u,v\}$ and $\{w,z\}$ are linearly independent.