Monotonic refinements of a Ky Fan inequality (Q2740874)

The author continues and extends earlier results [see Southeast Asian Bull. Math. 24, No. 3, 355-364 (2000; Zbl 0991.26013)] on continua of inequalities beween the entries in the Ky Fan-Wang-Wang inequalities NEWLINE\[NEWLINE{\mathfrak H_n( \underline a; \underline w)\over\mathfrak H_n' ( \underline a; \underline w)}\leq {\mathfrak G_n( \underline a; \underline w)\over\mathfrak G_n' ( \underline a; \underline w)}\leq {\mathfrak A_n( \underline a; \underline w)\over\mathfrak A_n' ( \underline a; \underline w)},NEWLINE\]NEWLINE where the notation \(\mathfrak H_n' ( \underline a; \underline w)\) for \(\mathfrak H_n' ( 1-\underline a; \underline w)\) is usual in this topic. Various functions that are continuous, increasing and either log-convex or log-concave are defined on the interval \([0, 1]\), functions that take the value \({\mathfrak H_n(\underline a; \underline w)\over\mathfrak H_n' ( \underline a; \underline w)}\), \({\mathfrak G_n( \underline a; \underline w)\over\mathfrak G_n' ( \underline a; \underline w)}\), at \(0\) and \({\mathfrak G_n( \underline a; \underline w)\over\mathfrak G_n' ( \underline a; \underline w)}\), \({\mathfrak A_n( \underline a; \underline w)\over\mathfrak A_n' ( \underline a; \underline w)}\), at \(1\). These generalizations of the author's earlier results depend on the properties of the simple function \(F( \lambda)= \prod_{i= 1}^n \bigl(\lambda a + (1- \lambda)a_i - k\bigr)^{w_i}\), where \(\underline a, \underline w \) are positive \(n\)-tuples, \(w_1 + \ldots + w_n = 1\), \(k< a_i, 1\leq i\leq n\), and \( a\geq{\mathfrak A_n( \underline a; \underline w)}\).