On some generalizations and refinements of a Ky Fan inequality (Q5929891)

This paper proves various extensions of the Ky Fan inequality and also gives an interesting proof, using Levinson 's inequality, of a previous extension by the same author [Southeast Asian Bull. Math. 22, No. 4, 363-372 (1998; Zbl 0947.26022)]. In particular we have the following results: for any \(k\geq 1\), NEWLINE\[NEWLINE(k-G_n')/(k-A_n')\leq A_n/G_nNEWLINE\]NEWLINE and if \(2G_n'\geq k\geq A_n'\) then NEWLINE\[NEWLINEA_n'/G_n'\leq (k-G_n')/(k-A_n');NEWLINE\]NEWLINE when \(k=1\) these inequalities reduce to results of \textit{H. Alzer} [Acta Appl. Math. 38, No. 3, 305-354 (1995; Zbl 0834.26013)], and Jiang; here \(A_n, G _n\) are the weighted arithmetic and geometric means of \(x_i, 0<x_i\leq 1/2, 1\leq i\leq n\), and \(A_n', G _n'\) are the weighted arithmetic and geometric means of \(1-x_i, 1\leq i\leq n\). A further result of \textit{G.-S. Yang} and \textit{C.-S. Wang} [J. Math. Anal. Appl. 201, No. 3, 955-965 (1996; Zbl 0854.26014)], is extended by showing that NEWLINE\[NEWLINE\prod_{i=1}^n\bigl(tz+(1-t)x_i\bigr)^{ \alpha_i}\biggl/\prod_{i=1}^n\bigl(t(1-z)+(1-t)(1-x_i)\bigr)^{ \alpha_i}NEWLINE\]NEWLINE is strictly increasing unless \(x_1=\cdots=x_n\); here \(0<x_i\leq 1/2, 1\leq i\leq n\), the weights \(\alpha_i\) are positive with \(\alpha_1+ \cdots+{\alpha_n = }1\), \(A_n\leq z\leq 1/2\); the results of Yang and Wang occur by taking equal weights and \(z=A_n\). The author then proves that other functions, introduced by himself and by Yang and Wang, that give continua of inequalities between the arithmetic and geometric mean can be deduced from this result.