On some refinements and converses of multidimensional Hilbert-type inequalities (Q2891967)

The author studies some new refinements and converses of multidimensional Hilbert inequalities with non-conjugate exponents. These results, that are too complicatd to quote here, are based on various interesting improvements of the classical Hölder inequality. One such result is the following: let \((\Omega, \Sigma,\mu)\) be a \(\sigma\)-finite measure space, \(F_i:\Omega\mapsto \mathbb R\) non-negative measurable functions, \(1\leq i\leq n\), \(\sum_{i=1}^n \alpha_i=1, A=\max \alpha_i, a=\min \alpha_i, \alpha _i>0, 1\leq i\leq n\); then NEWLINE\[NEWLINE \begin{multlined} na\prod_{i=1}^n\| F_i^{\alpha_i}\|_{1/ \alpha_i}\Biggl(1-{{\int_{\Omega}\prod_{i=1}^n F_i^{1/n}(x)d \mu(x)}\over{\prod_{i=1}^n\| F_i^{1/n}\|_n}}\Biggr) \leq \prod_{i=1}^n\| F_i^{\alpha_i}\|_{1/ \alpha_i}-\int_{\Omega}\prod_{i=1}^n F_i^{\alpha_i}(x)d \mu(x)\\ \leq nA\prod_{i=1}^n\| F_i^{\alpha_i}\|_{1/ \alpha_i}\Biggl(1-{{\int_{\Omega}\prod_{i=1}^n F_i^{1/n}(x)d \mu(x)}\over{\prod_{i=1}^n\| F_i^{1/n}\|_n}}\Biggr).\end{multlined} NEWLINE\]NEWLINE This inequality clearly gives a refinement of both Hölder's inequality and its converse.