A bilinear inequality
The author establishes that for an \(n\times n\) real symmetric matrix, \(X\neq 0\), the following inequality for the symmetric bilinear form \(a^{T}Xb\) holds: \[ (a^{T}Xb)^{2}\leq \frac{|x_{+}|_{\infty }^{2}+|x_{-}|_{\infty }^{2}}{2} (|a|^{2}|b|^{2}+(a,b)^{2}), \] for all \(a,b\in \mathbb{R}^{n}\), where \(|x_{+}|_{\infty }\) and \( |x_{-}|_{\infty }\) are the largest nonnegative eigenvalues of \(X\) and \(-X\), respectively, and \((a,b)=a^{T}b\) is the Euclidean inner product of \(a\) and \(b \). If \(X=I\), that inequality is equivalent to Cauchy-Schwarz inequality \( |(a,b)|\leq |a||b|\). If \(X\neq I\) and \(X\) is semi-positive definite, the inequality is strictly sharper than \(|a^{T}Xb|\leq |x|_{\infty }|a||b|\). However, when \(X\) is indefinite, for example \(X=\text{diag}(1,-1)\), the inequality might be weaker. The inequality is used to obtain inequalities in real separable Hilbert spaces, in particular, integral inequalities in \(L^{2}(\mathbb{R})\) and \( l^{2}(\mathbb{R})\). The inequality applied to the Hilbert matrix \((1/(i+j-1)) \), that following \textit{W. Magnus} [Am. J. Math. 72, 699-704 (1950; Zbl 0041.23805)] is a nonnegative definite operator on \(l^{2}(\mathbb{R})\) with spectral set \([0,\pi ]\), is the inner product version of Hilbert's inequality that is sharper than the Hilbert inequality. The author also deduces Schur's form of Hilbert's inequality in \(l^{2}(\mathbb{R})\) and the integral form of the inner product version of Hilbert's inequality in \(L^{2}(\mathbb{R })\).
scientific article; zbMATH DE number 1836754
scientific article