全平面上关联指数函数的半离散Hilbert型不等式

A Half-discrete Hilbert-type Inequality Related to Exponential Function in the Whole Plane

通过引入若干个参数,定义了一个全平面上与指数函数关联的核函数。借助实分析的技巧,特别是权系数的方法,建立了一个全平面上半离散的Hilbert型不等式。通过构造特殊的序列和函数,证明得出所建立的半离散Hilbert型不等式的常数因子为最佳值。文末对核函数进行特殊化,并引入Catalan常数和Euler数,得到了若干有趣的特殊结果。

By the introduction of several parameters,a kernel function in the whole plane and related to exponen‐tial function is defined.With the help of techniques of real analysis,especially the method of weight coefficient,a half-discrete Hilbert-type inequality in the whole plane is established.By constructing a special sequence and a special function,it is proved that the constant factor of the newly established inequality is the best possible.At the end of the paper,some special and interesting results are obtained by specifying the kernel function and introducing Catalan constant and Euler number.