Notes on the Schur-convexity of the extended mean values
Let \(f\) be a continuous function on an interval \(I\) and \(p\) be a continuous weight on \(I\).The weighted arithmetic mean of \(f\) is defined by \[ F(x,y)=\begin{cases} \frac{\int_{x}^{y}p(t)f(t)dt}{\int_{x}^{y}p(t)dt},&x\neq y \\ f(x),&x=y. \end{cases} \] The authors prove that \(F\) is Schur-convex (Schur-concave) on \(I^{2}\) if and only if the inequality \[ F(x,y)\leq \frac{p(x)f(x)+p(y)f(y)}{p(x)+p(y)} \] holds (reverses) for all \(x,y\in I.\) Schur-convexity of the extended mean values is also studied.
scientific article; zbMATH DE number 5001901