一个Hardy-Hilbert类不等式的一个改进

利用改进了的Cauchy不等式对1个类似于Hardy-Hilbert不等式的不等式作了改进.建立了1个新的不等式：〖ＤＤ（〗∞〖〗ｎ＝１〖ＤＤ）〗〖ＤＤ（〗∞〖〗ｍ＝１〖ＤＤ）〗〖ＳＸ（〗ａｍｂｎ〖〗ｌｎ　ｍ＋ｌｎ　ｎ＋１〖ＳＸ）〗＜π〖ＪＢ（｛〗〖ＤＤ（〗∞〖〗ｎ＝１〖ＤＤ）〗ｎａ２ｎ〖ＤＤ（〗∞〖〗ｎ＝１〖ＤＤ）〗ｎｂ２ｎ〖ＪＢ）｝〗１／２（１－Ｒ）１／２．其中Ｒ＝〖ＪＢ（（〗〖ＳＸ（〗（α，γ）〖〗‖α‖〖ＳＸ）〗－〖ＳＸ（〗（β，γ）〖〗‖β‖〖ＳＸ）〗〖ＪＢ））〗２.

An Improvement of the Inequality Similar to Hardy-Hilbert's Inequality

In this paper,by means of the refined Cauchy's inequality,a inequality similar to Hilbert's inequality is improved.A new inequality is built:∞∑n＝1∞∑m＝1(ambn)/(ln　m＋ln　n＋1)＜π∞∑n＝1na2n∞∑n＝1nb2n1／2(1－R)1／2．where R＝((α,γ))/(‖α‖)－((β,γ))/(‖β‖)2.