On the Equality of Bajraktarević Means to Quasi-Arithmetic Means

This paper offers a solution of the functional equation $$\begin{aligned}&\big (tf(x)+(1-t)f(y)\big )\varphi (tx+(1-t)y)\\&\quad =tf(x)\varphi (x)+(1-t)f(y)\varphi (y) \qquad (x,y\in I), \end{aligned}$$where $$t\in \,]0,1[\,$$, $$\varphi :I\rightarrow \mathbb {R}$$ is strictly monotone, and $$f:I\rightarrow \mathbb {R}$$ is an arbitrary unknown function. As an immediate application, we shed new light on the equality problem of Bajraktarević means with quasi-arithmetic means.