On Hilbert's Inequality and Its Applications

A Note on Hilbert Type Inequality

“…From_ (12) and (13) and using the Jensen's integral inequality (see [6]) we observe that For a number of generalizations, variants and extensions of the inequalities given in Theorems A and B, we refer the interested readers to [1][2][3][4][5]7,lOJ and the references given therein.…”

On the Hilbert Inequality With Weights

“…Further, in [5] it was shown that ω(n) < π − α √ n+1 where α = 0.5292496 + . The purpose of the present paper is to simplify and to refine the results of [4,5]. The method and theory employed by us are different from those in [4,5].…”

“…where the weight function ω is defined by ω(n) = π − θ(n) √ 2n + 1 . Recently, a few papers (see [4,5]) dealt with the weight function ω. Namely, in [4] it was shown that θ(n) > 4n+1 3(n+1)(2n+1) > 0 (n ∈ N 0 ). Clearly, this inequality is related to n, and 4n+1 3(n+1)(2n+1) → 0 as n → ∞.…”

“…The purpose of the present paper is to simplify and to refine the results of [4,5]. The method and theory employed by us are different from those in [4,5]. To be specific, we use the expansion of functions into power series and the approximation theory.…”

Inequalities Similar to the Integral Analogue of Hilbert's Inequality

“…Since the discovery of the most celebrated Hilbert's double series theorem and its integral analogue given in Theorem H, a large number of papers dealing with different proofs, numberous variants, various generalizations and their applications have appeared in the literature, see [1][2][3][4][5][6][7]12]. For the earlier historical development of this kind of inequalities and many important applications in analysis, see (3,Chapter IX).…”