A half-discrete Hilbert-type inequality in the whole plane with the constant factor related to a cotangent function

Abstract In this work, by the introduction of some parameters, a new half-discrete kernel function in whole plane is defined, which involves both homogeneous and nonhomogeneous cases. By employing techniques real analysis, especially method weight function, Hilbert-type inequality with as well its equivalent Hardy-type inequalities are established. Moreover, it proved that constant factors newly obtained best possible. Finally, assigning special values to kernels presented at end paper.