Hilbert-type inequalities for Hilbert space operators

Abstract

In this paper we establish a general form of the Hilbert inequality for positive invertible operators on a Hilbert space. Special emphasis is given to such inequalities with homogeneous kernels. In some general cases the best possible constant factors are also derived. Finally, we obtain the improvement of previously deduced results, based on the application of the Hermite-Hadamard inequality.

Keywords: Hilbert operator inequality, Holder operator inequality, Hermite-Hadamard inequality, Hilbert space, positive operator, geometric mean, homogeneous kernel, Beta function

Quaestiones Mathematicae 36(2013), 209-223