Extensions of some inequalities (Q5952864)

The extended mean values \(E(r,s;x,y)=\Big[\frac{r}{s}\cdot\frac{y^s-x^s} { y^r-x^r}\Big]^{{1/(s-r)}}\) were defined in [6] by K. B. Stolarsky in 1975. As a special case, we have \(E(1,2;x,y)=\frac{x+y}2=A(x,y)\), the arithmetic mean. It was verified that the extended mean values \(E(r,s;x,y)\) are increasing strictly with both \(r\) and \(s\) and with both \(x\) and \(y\). The inequalities, obtained in this paper, and those, obtained by Zh. Liu in [1], can be deduced easily from monotonicity of the extended mean values \(E(r,s;x,y)\). The convexities of the extended mean values \(E(r,s;x,y)\) have been considered by F. Qi. For more information, please refer to the following references. [1] \textit{Zh. Liu}, Remark on refinements and extensions of an inequality, J. Math. Anal. Appl. 234, No.~2, 529--533 (1999; Zbl 0932.26012). [2] \textit{F. Qi}, Logarithmic convexity of extended mean values, Proc. Am. Math. Soc. 130, No.~6, 1787--1796 (2002; Zbl 0993.26012). RGMIA Res. Rep. Coll. 2, No.~5, Art.~5, 643--652 (1999). Available online at \url{http://rgmia.vu.edu.au/v2n5.html}. [3] \textit{F. Qi}, \textit{Schur-convexity of the extended mean values}, Rocky Mountain J. Math., accepted. RGMIA Res. Rep. Coll. 4, No.~4, Art.~4, 529--533 (2001). Available online at \url{http://rgmia.vu.edu.au/v4n4.html}. [4] \textit{F. Qi}, The extended mean values: definition, properties, monotonicities, comparison, convexities, generalizations, and applications, RGMIA Res. Rep. Coll. 5, No.~1, Art.~5 (2002). Available online at \url{http://rgmia.vu.edu.au/v5n1.html}. [5] \textit{F. Qi, J. Sándor, S. S. Dragomir}, and \textit{A. Sofo}, Notes on the Schur-convexity of the extended mean values, RGMIA Res. Rep. Coll. 5, No.~1, Art.~3 (2002). Available online at \url{http://rgmia.vu.edu.au/v5n1.html}. [6] \textit{K. B. Stolarsky}, Generalizations of the logarithmic mean, Mag. Math. 48, 87--92 (1975; Zbl 0302.26003).