A Hilbert-Type Integral Inequality in the Whole Plane Related to the Exponential Function

A new parametric Hilbert’s integral inequality in the whole plane is discussed. Two weight functions are established, and the two weight functions are calculated as constants by using the integral variable substitution method. By using the weighted H lder’s inequality, the appropriate formula, the real analysis techniques, and by introducing the non-zero independent parameters and the parameters satisfying , a new Hilbert-type integral inequality with intermediate variable, the best constant factor and non-homogeneous kernel are obtained, as well as its equivalent inequality and the inequalities with homogeneous or non-homogeneous kernel under special parameters. The result shows that this new parametric Hilbert’s integral inequality in the whole plane is superior, and is the expansion of the past results. The study of Hilbert’s integral inequality extends from the first quadrant to the whole plane in this paper, which provides a broader idea for the depth study of Hilbert’s inequality, enriches and perfects the study of Hilbert’s inequality system.