On the invariance equation for two-variable weighted nonsymmetric Bajraktarevi\'c means

The purpose of this paper is to investigate the invariance of the arithmetic mean with respect to two weighted Bajraktarevi\'c means, i.e., to solve the functional equation $$\bigg(\frac{f}{g}\bigg)^{\!\!-1}\!\!\bigg(\frac{tf(x)+sf(y)}{tg(x)+sg(y)}\bigg) +\bigg(\frac{h}{k}\bigg)^{\!\!-1}\!\!\bigg(\frac{sh(x)+th(y)}{sk(x)+tk(y)}\bigg)=x+y \qquad(x,y\in I),$$ where $f,g,h,k:I\to\mathbb{R}$ are unknown continuous functions such that $g,k$ are nowhere zero on $I$, the ratio functions $f/g$, $h/k$ are strictly monotone on $I$, and $t,s\in\mathbb{R}_+$ are constants different from each other. By the main result of this paper, the solutions of the above invariance equation can be expressed either in terms of hyperbolic functions or in terms of trigonometric functions and an additional weight function. For the necessity part of this result, we will assume that $f,g,h,k:I\to\mathbb{R}$ are four times continuously differentiable