Abstract
By using real analysis and weight functions, we obtain a few equivalent statements of a Hilbert-type integral inequality in the whole plane related to the kernel of exponent function with intermediate variables. The constant factor related to the gamma function is proved to be the best possible. We also consider some particular cases and the operator expressions.
Highlights
If 0 < ∞ 0 f 2(x) dx < ∞ and ∞ 0 g2(y) dy∞, we have the following well-knownHilbert integral inequality:∞ ∞ f (x)g(y) dx dy < π
The constant factor related to the gamma function is proved to be the best possible
When statement (iii) holds, namely, Kα(γ,β)(σ ) ≤ M, if there exists a constant M(≤ Kα(γ,β)(σ )) such that (26) is valid, M = Kα(γ,β)(σ ), and we can conclude that the constant factor M = Kα(γ,β)(σ ) in (26) is the best possible
Summary
Hilbert integral inequality (see [1]):. 0 0 x+y where the constant factor π is the best possible. He et al [11,12,13,14,15,16,17,18,19] proved some new Hilbert-type integral inequalities in the whole plane with the best possible constant factors. Lemma 5 We define the following weight functions: ωδ(σ , y) := yσβ e–ρ(xδαyβ )γ xδασ –1 dx (y ∈ R),. Where Kα(γ,β)(σ ) is the best possible constant factor; (2) for δ = –1, M = Kα(γ,β)(σ ), we have the following equivalent inequalities with homogeneous kernel of degree 0:. When statement (iii) holds, namely, Kα(γ,β)(σ ) ≤ M, if there exists a constant M(≤ Kα(γ,β)(σ )) such that (26) is valid, M = Kα(γ,β)(σ ), and we can conclude that the constant factor M = Kα(γ,β)(σ ) in (26) is the best possible.
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