Some refinements of Ky Fan's and Sándor's inequalities (Q1885191)

By using some refinements of Jensen's discrete inequality for convex functions, the author proves refinements of the Ky Fan inequality \(A/G\geq A'/G'\), resp. Sándor's inequality \(1/A- 1/A'\leq 1/H- 1/H'\), where \(A\), \(G\), \(H\) denote the arithmetic, geometric, resp. harmonic means of numbers \(x_i\in (0,{1\over 2}]\) \((i= 1,\dots, n)\), and \(A'= A'(X)= A(1- X)\), etc., with \(X= (x_i)\). \{Reviewer's remark: Inequality (1.4) was discovered by the reviewer in [Aequationes Math. 40, No. 2/3, 261--270 (1990; Zbl 0717.26014)], while relation (27), in a more general form, by \textit{E. Neuman} and the reviewer [Math. Inequal. Appl. 5, No. 1, 49--56 (2002; Zbl 1007.26015)]\}.