Abstract
By the use of the methods of real analysis and the weight functions, a few equivalent conditions of a Hilbert-type integral inequality with the nonhomogeneous kernel in the whole plane are obtained. The best possible constant factor is related to the extended Riemann zeta function. As applications, a few equivalent conditions of a Hilbert-type integral inequality with the homogeneous kernel in the whole plane are deduced. We also consider the operator expressions and a few particular cases.
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