Abstract
Using weight coefficients, a complex integral formula, and Hermite–Hadamard’s inequality, we give an extended reverse Hardy–Hilbert’s inequality in the whole plane with multiparameters and a best possible constant factor. Equivalent forms and a few particular cases are considered.
Highlights
IntroductionN=1 m=1 where the constant factor B(λ1, λ2) is the best possible, and B(u, v) is the beta function defined as (see [6])
If p > 1 p + 1 q = 1, am, bn ≥
Since cs > · · · > c1 = c1 > 0, by (8) we find ks(λ1) :=
Summary
N=1 m=1 where the constant factor B(λ1, λ2) is the best possible, and B(u, v) is the beta function defined as (see [6]). Hadamard’s inequality, we give the following extension of the reverse of (1) in the whole plane: If We prove an extended inequality of (6) with multiparameters and a best possible constant factor. Lemma 2 With regards to the above agreement, replacing 0 < λ1 ≤ 1 (0 < λ1 < 1) by λ1 > 0 and setting hβ (λ1) := 2ks(λ1) csc β, we still have hβ (λ1) 1 – θ (λ2, m) < ω(λ2, m) < hβ (λ1), |m| ∈ N,. Lemma 3 With regards to the above agreement, replacing 0 < λ2 ≤ 1 (0 < λ2 < 1) by λ2 > 0, for hα(λ1) = 2ks(λ1) csc α, we still have hα(λ1) 1 – θ(λ1, n) < (λ1, n) < hα(λ1), |n| ∈ N,.
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