Some improvements on Hilbert's integral inequality
The main idea of this paper is that by expanding a Gram determinant of order three, with one element having norm 1, one can obtain a refinement of the classical Cauchy-Schwarz inequality in the form \[ \| a\|^2\cdot\| b\|^2- |(a,b)|^2- G(a,b,c)\geq 0, \] where \(\| c\|= 1\). Then for the space \(L^2(0,\infty)\) with scalar product \((f,g)= \int^\infty_0 f(x)\overline{g(x)}dx\) one can select a proper definite quadratic form \(F(x(f, g),y(f,g))\) such that the right-hand side of the Hilbert inequality can be replaced by \[ \pi^2\cdot \int^\infty_0 f^2(t)dt \int^\infty_0 g^2(s)ds- F(x(f, g),y(f, g)). \] Other related inequalities, as Widder's or Hardy-Littlewood's are improved by the same manner. Unfortunately, there appear certain misprints in this paper.
scientific article; zbMATH DE number 1251439