On the Equality of Bajraktarević Means to Quasi-Arithmetic Means

This paper offers a solution of the functional equation (tf(x)+(1-t)f(y))φ(tx+(1-t)y)=tf(x)φ(x)+(1-t)f(y)φ(y)(x,y∈I),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\big (tf(x)+(1-t)f(y)\big )\varphi (tx+(1-t)y)\\&\quad =tf(x)\varphi (x)+(1-t)f(y)\varphi (y) \qquad (x,y\in I), \end{aligned}$$\end{document}where t∈]0,1[\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\in \,]0,1[\,$$\end{document}, φ:I→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi :I\rightarrow \mathbb {R}$$\end{document} is strictly monotone, and f:I→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:I\rightarrow \mathbb {R}$$\end{document} is an arbitrary unknown function. As an immediate application, we shed new light on the equality problem of Bajraktarević means with quasi-arithmetic means.