A simple proof of monotonicity for extended mean values (Q1270915)

\textit{K. B. Stolarsky} [Math. Mag. 48, 87-92 (1975; Zbl 0302.26003)] introduced the mean values given, for positive \(x,y\), by \[ E(r,s;x,y) = [r(y^{s}-x^{s})]^{1/(s-r)} [s(y^{r}-x^{r})]^{1/(r-s)} \] and their limits as \(r\rightarrow 0,\;s\rightarrow 0,\;r\rightarrow s\) or \(x\rightarrow y\). The authors offer a ``simple proof'' of the result by \textit{E. B. Leach} and \textit{M. C. Sholander} [J. Math. Anal. Appl. 92, 207-223 (1983; Zbl 0517.26007)] which states that \(E\) increases in both parameters \(r,s\) and in both variables \(x,y\). [The proof of monotonicity in \(x\) and \(y\) could be more detailed].