On the equality of Bajraktarevi\'c means to quasi-arithmetic means

This paper offers a solution of the functional equation $$(tf(x)+(1-t)f(y))\phi (tx+(1-t)y)=tf(x)\phi (x)+(1-t)f(y)\phi (y)(x,y\in I),$$ where $t\in ]0,1[$ is a fixed number, $\phi :I\to \mathbb{R}$ is strictly monotone, and $f:I\to \mathbb{R}$ is an arbitrary unknown function. As an immediate application, we shed new light on the equality problem of Bajraktarevi'c means with quasi-arithmetic means.