Über Lehmers Mittelwertfamilie. (On Lehmer's family of means) (Q919472)

The means \(L_ r(x,y)=(x^{r+1}+y^{r+1})/(x^ r+y^ r)\) and their multivariable analogues are attributed to \textit{D. H. Lehmer} [J. Math. Anal. Appl. 36, 183-200 (1971; Zbl 0222.26018)] even though they go back at least to \textit{K. Goldziher} [Allg. Statist. Archiv 10, 307-313 (1917); Skand. Aktuarietidskr. 1920, 72-96 (1920)] and were object of systematic research by \textit{E. F. Beckenbach} [Am. Math. Mon. 57, 1-6 (1950; Zbl 0035.157)]. Here inequalities like \[ (r-s)L_ t(x,y)+(s-t)L_ r(x,y)+(t-r)L_ s(x,y)\geq 0\text{ for } 0\leq r<s<t, \] \[ (t-r)(t-s)L_ t(x,y)+(r-s)(r-t)L_ r(x,y)+(s-t)L_ s(x,y)\geq 0 \] are offered for positive x,y.