We first establish a Hilbert-type inequality and its equivalent Hardy form with best possible constant factors by constructing a homogeneous kernel function involving hyperbolic functions. Furthermore, we introduce Bernoulli numbers and the partial fraction expansions of trigonometric functions, and then we present several special and interesting Hilbert-type inequalities, in which the constant factors are represented by Bernoulli numbers and by some higher derivatives of trigonometric functions.
"On a Hilbert-type inequality with homogeneous kernel involving hyperbolic functions." Rocky Mountain J. Math. 50 (5) 1871 - 1881, October 2020. https://doi.org/10.1216/rmj.2020.50.1871