General Hilbert's and Hardy's inequalities

Abstract

Abstract. In this paper we make some further generalizations of well known Hilbert's inequality and its equivalent form in two dimensional case. We also derive some results on Hardy's inequality. Then we apply our general results to homogeneous functions. A reverses of Hilbert's inequality are also given in integral case. Many other results of this type in recent years, follows as a special case of our results. (2000): 26D15. Mathematics subject classification

A Hilbert inequality and an Euler-Maclaurin summation formula

“…Further, by putting k = 1 in Theorem 3.1 we obtain the following result. In such a way improve our results from [6] and [5], as well as some previously known results which include the beta function.…”

“…One possibility of generalizing it is by extending it to the case of non-conjugate parameters. We shall also compare our results with some previously known from the literature and obtain some improvements of our results from [6] and [5]. [3] A Hilbert inequality and an Euler-Maclaurin summation formula 421…”

A New Reverse Hilbert's Type Inequality

“…In [2], [9] and [6], M. Krnic, J. Pecaric and B. Yang gave some generalization and reinforcement of inequality (1.1). In [4], Kuang Jichang and L. Debnath gave a reinforcement of inequality (1.2):…”

A Strengthened Form of a General Hardy-Type Inequality Obtained via Superquadraticity and Its Applications

“…In paper [14], the mentioned equivalence was established in a more general manner. More precisely, paper [14] provides a unified treatment of the Hilbert-type and the Hardy-Hilbert-type inequalities with conjugate exponents. We refer here only to the Hardy-Hilbert-type inequality involving a general non-negative homogeneous kernel k of degree −s, s > 0.…”