Chu, Yuming; Zhang, Xiaoming; Wang, Gendi The Schur geometrical convexity of the extended mean values. (English) Zbl 1163.26004 J. Convex Anal. 15, No. 4, 707-718 (2008). The extended mean value \(E(r,s;x,y)\) is a function of \(r,s\in \mathbb{R}\) and \(x,y>0\) which includes, for various values of \(r\) and \(s\), many well-known mean values of two positive numbers \(x\), \(y\): arithmetic mean, geometric mean, harmonic mean, Hölder mean, logarithmic mean etc. The main result of the paper is that \(E(r,s;\cdot,\cdot)\) is Schur geometrically convex (resp., concave) on \((0,+\infty)\times(0,\infty)\) if and only if \(s+r\geq0\) (resp., \(r+s\leq0\)). Reviewer: Nicolas Hadjisavvas (Hermoupolis) Cited in 32 Documents MSC: 26B25 Convexity of real functions of several variables, generalizations Keywords:extended mean value; Schur convex; Schur concave; Schur geometrically convex; Schur geometrically concave PDFBibTeX XMLCite \textit{Y. Chu} et al., J. Convex Anal. 15, No. 4, 707--718 (2008; Zbl 1163.26004) Full Text: Link