Inequalities Similar to the Integral Analogue of Hilbert's Inequality

Abstract


 
 
 In the present paper we establish some new inequalities similar to the integral ana­logue of Hilbert's inequality by using a fairly elementary analysis. 
 
 

Generalization of dynamic Hilbert-Pachpatte inequality via Taylor formula on time scale

“…In 1908, Weyl [3] presented a proof of the Hilbert inequality (1). In 1911, Schur [4] proved a Hilbert integral analogue of…”

“…This proves the theorem.Remark 21 In Theorem 20, if T = R; a = 0 and p = q = 2, then we have[1, Theorem 3].Remark 22 In Theorem 20, if T = R; a = 0; p = q = 2 and k = 0, then we have [1, Remark 1, inequality (5 8 )].Corollary 23 Under the hypotheses of Theorem 20, if T = N and a = 1 we get x ) r k ! ( )…”

On some inequalities involving the function and its fractional derivative

“…Another inequality that interests us in this paper is the Hilbert‐type inequality, established by Pachpatte 13 . This inequality can be stated as follows: Let n ≥ 1 and 0 ≤ k ≤ n − 1 be integers.…”

“…Another inequality that interests us in this paper is the Hilbert-type inequality, established by Pachpatte. 13 This inequality can be stated as follows: Let n ≥ 1 and 0 ≤ k ≤ n À 1 be integers. Let u ∈ C n ([0, x]) and v ∈ C n ([0, y]), where x > 0, y > 0, and let u ðiÞ ð0Þ ¼ v ðjÞ ð0Þ ¼ 0, j∈f0, 1, …,n À 1g.…”