On the best constant in Hilbert's inequality

Abstract

Abstract. The main objective of this paper is a study of some new generalizations of Hilbert's type inequalities. More precisely, we obtain, in some general cases, that the constants involved in the right-hand sides of such inequalities are the best possible. (2000): 26D15. Mathematics subject classification

Extension of Hilbert's inequality

“…Second inequality (13) and the equivalence of the inequalities (12) and (13) is shown in the same way as in the paper [3] (see Theorem 5). 2 In our paper [4] we have obtained the best possible constants for some special choices of parameters A 1 and A 2 . We can also obtain the best possible constants for previous extension of Hilbert's inequality.…”

Discrete Hilbert-type inequalities with general homogeneous kernels

“…More precisely, in [4], the authors have obtained the best possible constant factors in the Hilbert and Hardy-Hilbert type inequalities, with homogeneous kernel K(x, y) = (x + y) −λ , λ > 0, and the functions xu(x) and yv(y), respectively instead of u(x) and v(x).…”

On a strengthened multidimensional Hilbert-type inequality

“…In papers [1], [2], [5], [9], the authors investigated the conditions under which the constant factors involved in appropriate Hilbert-type inequalities are the best possible in the sense that they can not be replaced with the smaller constants.…”

“…Moreover, the parameters A ij defined by (4.5), can satisfy the set of conditions as in (3.6) only for n = 2. In this case, the set of conditions (3.6) in (4.6) for n = 2 (see also paper [5]). …”