Measurability of \((M,N)\)-Wright convex functions
Let \(I\subset\mathbb R\) be an open interval. A function \(M:I^2\rightarrow I\) is called a mean if \[ \min\{x,y\}\leq M(x,y)\leq \max\{x,y\},\quad x,y\in I. \] A function \(f:I\rightarrow\mathbb R\) is said to be \((M,N)\)-Wright convex if \[ f(M(x,y))+f(N(x,y))\leq f(x)+f(y),\quad x,y\in I, \] where \(M\) and \(N\) are means such that \[ M(x,y)+N(x,y)=x+y,\quad x,y\in I. \] The author gives a general criterion for \(M\) and \(N\) for which every \((M,N)\)-Wright convex, Lebesgue measurable function is convex. Let \(\varphi:I\rightarrow\mathbb R\) be a solution of the functional equation \[ \varphi(M(x,y))+\varphi(N(x,y))=\varphi(x)+\varphi(y),\quad x,y\in I. \] Sufficient conditions on \(M,N\) and \(\varphi\) are provided such that for every Lebesgue measurable, \((M,N)\)-Wright convex function \(f\), the function \(f\circ\varphi^{-1}\) is convex.
scientific article; zbMATH DE number 5843955