On a more accurate half-discrete Hilbert-type inequality involving hyperbolic functions

Abstract

In this work, by the introduction of a new kernel function composed of exponent functions with several parameters, and using the method of weight coefficient, Hermite-Hadamard’s inequality, and some other techniques of real analysis, a more accurate half-discrete Hilbert-type inequality including both the homogeneous and non-homogeneous cases is established. Furthermore, by introducing the Bernoulli number and the rational fraction expansion of tangent function, some special and interesting Hilbert-type inequalities and their equivalent hardy-type inequalities are presented at the end of the paper.

Keywords

• hilbert-type inequality
• hyperbolic function
• hermite-hadamard’s inequality
• rational fraction expansion
• bernoulli number
• 26d15
• 41a17