In a generalization of the classical Hilbert inequality by Hardy, Littlewood and Pólya, the best constant for an inequality is determined provided that the generating function for the corresponding matrix satisfies certain monotonicity condition. In this paper, we determine the best constant for a class of inequalities when the monotonicity condition is no longer satisfied. Mathematics subject ...

This is a survey on reverse Carleson measures for various Hilbert spaces of analytic functions. These spaces include the Hardy, Bergman, certain harmonically weighted Dirichlet, Paley-Wiener, Fock, model (backward shift invariant), and de Branges-Rovnyak spaces. The reverse Carleson measure for backward shift invariant subspaces in the non-Hilbert situation is new.

We formulate and prove a generalization of Hardy’s inequality [27] in terms random variables show that it contains the usual (or familiar) continuous discrete forms inequality. Next we improve recent version by Li Mao [42] with weights for general Borel measures mixed norms so implies Liao [43] Hardy Muckenhoupt [48] as well norm versions due to Littlewood [29], Bliss [8], Bradley [14]. An equi...

We explore a variety of pleasing connections between analysis, number theory and operator theory, while revisiting a number of beautiful inequalities originating with Hilbert, Hardy and others. We shall first establish the aforementioned Hilbert inequality [?, ?] and then apply it to various multiple zeta values. In consequence we obtain the norm of the classical Hilbert matrix, while illustrat...

This Ph.D. thesis deals with various generalizations of the inequalities by Carleman, Hardy and Pólya-Knopp. In Chapter 1 we give an introduction and overview of the area that serves as a frame for the rest of the thesis. In Chapter 2 we consider Carleman’s inequality, which may be regarded as a discrete version of Pólya-Knopp’s inequality and also as a natural limiting inequality of the discre...

In this paper, by using the Euler-Maclaurin expansion for the zeta function and estimating the weight function effectively, we derive a strengthenment of a Hardy-Hilbert’s type inequality proved by W.Y. Zhong. As applications, some particular results are considered.