By means of the techniques of the real analysis and the weight functions, a few
equivalent conditions of a Hilbert-type integral inequality with the general
nonhomogeneous kernel in the whole plane are obtained. The constant factor
is proved to be the best possible. As applications, a few equivalent
conditions of a Hilbert-type integral inequality with the general
homogeneous kernel in the whole plane are deduced. We also consider the
operator expressions, a few particular cases and some examples.