An extension of Ky Fan inequality (Q2782768)

Let \((X, \mu)\) be a measure space, \(0< \mu< \infty\), and let \(\mathfrak A_X(f; \mu), \mathfrak G_X(f; \mu)\) be the arithmetic, geometric mean, respectively, of the positive measurable function \(f\) that has co-domain \(]0,1/2]\); and let \(Y\subseteq X\) be \(\mu\)-measurable with \(\mu Y>0 \). Further define NEWLINE\[NEWLINE {\mathcal A}_X(f; \mu) ={\mathfrak A_X(f; \mu)\over\mathfrak A_X(1-f; \mu)},NEWLINE\]NEWLINE and analogously \({\mathcal G}_X(f; \mu)\). The main results of the author are: NEWLINE\[NEWLINE\biggl({{{\mathcal A}_Y(f; \mu)}\over {{\mathcal G}_Y(f; \mu)}}\biggr)^{\mu Y}\leq \biggl( {{{\mathcal A}_X(f; \mu)}\over{ {\mathcal G}_X(f; \mu)}}\biggr)^{\mu X}\tag{P}NEWLINE\]NEWLINE with equality if and only if \(f(x)= \mathfrak A_Y(f; \mu)\), \(\mu\)-almost everywhere on \(X\setminus Y\); NEWLINE\[NEWLINE\mu X \biggl(\mathfrak A_X(f; \mu)-{\mathfrak G_X(f; \mu)\over{\mathfrak G_X(f; \mu)+\mathfrak G_X(1-f; \mu)}}\biggr)\geq\mu Y \biggl(\mathfrak A_Y(f; \mu)-{\mathfrak G_Y(f; \mu)\over{\mathfrak G_Y(f; \mu)+\mathfrak G_Y(1-f; \mu)}}\biggr) \tag{R}NEWLINE\]NEWLINE with equality if and only if \(f(x) = {\mathfrak G_Y(f; \mu)\over{\mathfrak G_Y(f; \mu)+\mathfrak G_Y(1-f; \mu)}} \mu\)-almost everywhere on \(X\setminus Y\). (R) is a Rado-type extension, (P) a Popoviciu-type extension, of the integral form of Ky Fan's inequality; taking \(X=\{1,\ldots ,n\}\), \(Y=\{1,\ldots ,n-1\}\), \(\mu\) counting measure gives the results for the discrete form of Ky Fan's inequality. The proofs use the concavity of the function \(\log\bigl( x/(1-x)\bigr)\) on \(]0,1/2]\) . There are several misprints in the paper.