一个Hilbert型不等式及其等价形式的加强推广

ON A STRENGTHENED AND GENERALIZED HILBERT-TYPE INEQUALITY AND ITS EQUIVALENT FORM

本文研究了Hilbert型级数不等式.利用改进的Euler-Maclaurin求和公式及权系数的方法,并引进两对共轭指数,获得一个具有最佳常数系数的Hilbert型不等式,该不等式是联系若干Hilbert型不等式的加强推广式.作为应用,给出该不等式的等价形式及若干有意义的特殊结果.

In this article,we study the Hilbert-type inequality.By using the improved Euler-Maclaurin＇s summation formula and estimating the weight coefficient and introducing two pairs of indexes of conjugate,we get a Hilbert-type inequality with a best constant coefficient which is a strengthened and generalized form of some early Hilbert-type inequalities.As applications,we also consider the equivalent form of this inequality and deal with some particulars.