By means of the technique of real analysis and the weight functions, a few equivalent statements of a Hilbert-type integral inequality with the nonhomogeneous kernel in the whole plane are obtained. The constant factor related the beta function is proved to be the best possible. As applications, the case of the homogeneous kernel, the operator expressions and a few corollaries are considered.
Department of Mathematics, Guangdong University and of Education, Guangzhou, Guangdong 510303, P. R. China.
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