Hardy-Hilbert-Type Inequalities with a Homogeneous Kernel in Discrete Case

Hilbert and Hardy-Hilbert type inequalities see 1 are very significant weight inequalities which play an important role in many fields of mathematics. Although classical, such inequalities have attracted the interest of numerous mathematicians and have been generalized in many different ways. Also the numerous mathematicians reproved them using various techniques. Some possibilities of generalizing such inequalities are, for example, various choices of nonnegative measures, kernels, sets of integration, extension to multidimensional case, and so forth. Similar inequalities, in operator form, appear in harmonic analysis where one investigates properties of boundedness of such operators. This is the reason why Hilbert’s inequality is so popular and represents field of interest of numerous mathematicians: since Hilbert till nowadays. We start with the following two discrete inequalities, which are the well-known Hilbert and Hardy-Hilbert type inequalities. More precisely, if p > 1, 1/p 1/q 1, an, bn ≥ 0, such that 0 < ∑∞ n 0 a p n < ∞ and 0 < ∑∞ n 0 b q n < ∞, then the following inequality holds Hardy et al. 1 :