On Ky Fan's inequality (Q2720283)

The authors prove the Ky-Fan like inequalities \({\mathfrak H_n( \underline a; \underline w)\over\mathfrak H_n' ( \underline a; \underline w)}\leq {\mathfrak G_n( \underline a; \underline w)\over\mathfrak G_n' ( \underline a; \underline w)}\), where the notation \(\mathfrak H_n' ( \underline a; \underline w)\) for \(\mathfrak H_n' ( 1-\underline a; \underline w)\) is usual in this topic. In addition they prove a similar inequality with the harmonic, geometric means replaced by the logarithmic, identric means, respectively; the definition of these means for \(n\)-tuples being those of \textit{A. O. Pittenger} [Am. Math. Mon. 92, 99-104 (1985; Zbl 0597.26027)], and \textit{J. Sándor} and \textit{T. Trif} [Math. Inequal. Appl. 2, No. 4, 529-533 (1999; Zbl 0945.26020)]. The proof depends very simply on the convexity of the function \(\log(1-x) - \log x\) and a use of a theorem of \textit{M. L. Slater} [J. Approximation Theory 32, 160-166 (1981; Zbl 0521.41011)], the proof of which is unnecessarily included in the paper without attribution; it is available in the literature, for instance on pp. 63-64 of the book by \textit{J. E. Pečarić, F. Proschan} and \textit{Y. L. Tong} [``Convex functions, partial ordering, and statistical applications'' (1992; Zbl 0749.26004)]. Given the known chain of inequalities between these various means, NEWLINE\[NEWLINE\mathfrak H_n\leq \mathfrak G_n\leq \mathfrak L_n\leq \mathfrak I_n\leq \mathfrak A_n,NEWLINE\]NEWLINE it is then natural, given the classical Ky Fan-Wang-Wang inequalities to ask about the validity of \({\mathfrak L_n( \underline a; \underline w)\over\mathfrak L_n' ( \underline a; \underline w)}\leq {\mathfrak I_n( \underline a; \underline w)\over\mathfrak I_n' ( \underline a; \underline w)}\) . The authors point out that this cannot hold in general by providing samples of weights for which the inequality fails and for which the inequality and its reverse hold.