Equivalent integral conditions related to bilinear Hardy-type inequalities

Abstract

Infinitely many, even scales of, equivalent conditions are derived to characterize the bilinear Hardy-type inequality under various ranges of parameters.

On Bilinear Hardy Inequality and Corresponding Geometric Mean Inequality

“…Proof. The equivalence B1 (α, β, γ) ≈ Bi (α, β, γ, s 1 , s 2 ) ≈ Bj (α, β, γ, s) , (i = 2, 3, 10) , (j = 13, 14, 15) has been proved in [7]. Here, we give different proofs for the equivalence…”

“…The next aim of this paper is to prove a result of the type of Theorem A in respect of the bilinear Hardy inequality (1.5). Some of these equivalent conditions have recently been proved in [7]. In this paper, we give different proofs of some of the conditions proved in [7] and moreover, we provide several new equivalent conditions.…”

“…Some of these equivalent conditions have recently been proved in [7]. In this paper, we give different proofs of some of the conditions proved in [7] and moreover, we provide several new equivalent conditions. This is done in Section 3.…”