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A Hilbert-type integral inequality in the whole plane with the homogeneous kernel of degree \(-2\). (English) Zbl 1218.26031

Assume that \(f(x)\) and \(g(x)\) are non-negative and both of them satisfy \(0<\int_0^\infty f^2(x)\,dx<\infty\). Then, Hilbert’s integral inequality asserts that
\[ \int_0^\infty \int_0^\infty \frac{f(x)g(y)}{x+y}\,dx\,dy< \pi\left(\int_0^\infty f^2(x)\,dx\int_0^\infty g^2(x)\,dx\right)^\frac12, \]
where \(\pi\) is the best possible constant. The main object of the paper is to give a new Hilbert-type inequality in the whole plane with a homogeneous kernel of degree \(-2\) involving some parameters and a best constant factor.

MSC:

26D15 Inequalities for sums, series and integrals
26D10 Inequalities involving derivatives and differential and integral operators
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References:

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