Ky Fan inequality and bounds for differences of means (Q1864339)

Let \({\pmb\omega}=\{\omega_1,\dots, \omega_n\}\) be a positive \(n\)-tuple with \(\sum_{i=1}^n \omega_i = 1\) and \({\mathbf x}=\{x_1,\dots, x_n\}\) be a positive increasing \(n\)-tuple with \(x_n\leq 1\); then put \( {\mathbf x}'=\{1-x_1,\dots, 1-x_n\}\), denote by \( P_{n,r}\), respectively \( P_{n,r}{'}\), the \(r\)-th weighted power mean of \( {\mathbf x}\), respectively \( {\mathbf x}'\), and let \( \sigma_n = \sum_{i=1}^n \omega_i(x_i-P_{n,1})^2\). The following inequalities are considered: if \(r>s\) then (i) if \( {\mathbf x}\) is non-constant \[ {x_1\over 1-x_1} < { P_{n,r}^{'}- P_{n,s}^{'}\over P_{n,r} -P_{n,s}}<{x_n\over 1-x_n} \] and \[ {r-s\over 2x_1}\sigma_n \geq P_{n,r} -P_{n,s}\geq{r-s\over 2x_n}\sigma_n \tag{ii} \] In the case \(r=1, s=0\) (ii) is due to \textit{D. I. Cartwright} and \textit{M. J. Field} [Proc. Am. Math. Soc. 71, 36-38 (1978; Zbl 0392.26010)], and (i) has been deduced from the special case of (ii) by \textit{P. R. Mercer} [Math. Inequal. Appl. 3, No.~1, 147-148 (2000; Zbl 0945.26021)]. The main result of this paper is: for given \(r>s\) inequality (i) with \(x_n\leq 1/2\), inequality (i), and inequality (ii) are equivalent. Deductions are then made: (a) if (ii) holds then \(0\leq r+s\leq 3\); (b) if \(r=1\) then (ii) holds if and only if \(-1\leq s<1\); (c) if \(s=1\) then (ii) holds if and only if \(1<r\leq 2\). The case \(r=1, s=-1\) solves a problem of Mercer. In addition, the author proves an extenssion of (ii) that implies yet another extension of the Ky Fan inequality.