We first establish a new Hilbert-type inequality involving the kernel including both the homogeneous and the nonhomogeneous cases, and the constant factor of the newly obtained inequality is proved to be the best possible. Additionally, by means of Leibniz integral rule and the rational fraction expansion of trigonometric functions, we derive the representation of the best possible constant factor by the logarithmic function and the higher derivative of trigonometric functions, which leads to a few special and interesting inequalities. Furthermore, We also demonstrate that our newly obtained inequality is a unified generalization of some classical Hilbert-type inequalities.
"A unified extension of some classical Hilbert-type inequalities and applications." Rocky Mountain J. Math. 51 (5) 1865 - 1877, October 2021. https://doi.org/10.1216/rmj.2021.51.1865