A Hilbert-type inequality related to the higher derivative of cotangent function(一个关联余切函数高阶导数的Hilbert型不等式)

Abstract

By introducing multiple parameters, and using partial fraction expansion of Cotangent function, a new Hilbert-type inequality defined in the whole plane with the constant factor related to the higher derivative of cotangent function is established. Also, the equivalent form of the inequality is considered. Furthermore, assigning different values to the parameters, some special Hilbert-type inequalities defined in the whole plane are obtained.(通过引入多个参数，借助余切函数的部分分式展开式，在全平面上建立了最佳常数因子及与余切函数的高阶导数有关的Hilbert型不等式及其等价形式。特别地，通过对参数赋值，还给出了一些特殊的在全平面上的Hilbert型不等式。)

Keywords

• hilbert inequality(hilbert不等式)
• cotangent function(余切函数)
• partial fraction expansion(部分分式展开)
• hurwitz zeta function(hurwitz zeta函数)
• gamma function(gamma函数)