Hardy不等式的加强式及自动验证程序

A Sharpened Version of Hardy'S Inequality and the Automated Verification Program

研究Hardy不等式的加强式,通过对权系数W（k,p）的估计,在权系数W（k,2.5）下建立加强式∞∑n=1（1/nn∑k=1ak）^5/2≤25√15/27∞∑n=1[1-1/n^3/5（1-9√15/125W（1,2,5）]an^5/2编写程序hdiscover2012,实现了形如∞∑n=1（1/nn∑k=1ak）^p≤（p/p-1）^p∞∑n=1[1-1/n^1-1/p）^pW（q,p）]an^p加强式的自动验证,式中系数1-（p-1/p）~p W（1,p）为最佳.最后猜想上述不等式对p〉1成立.

The purpose of this paper is to investigate the sharpened version of Hardy＇s in- equality, by evaluating the weight coefficient W（k, p）, a sharp Hardy＇s inequality is established under the weight coefficient W（k, p）, as follows：∞∑n=1（1/nn∑k=1ak）^p≤（p/p-1）^p∞∑n=1[1-1/n^1-1/p）^pW（q,p）]an^p,where the constant 1 -（p-1/p）^pW（1,p） is the best possible. At the end of this paper aconjecture related to the above inequality for p 〉 1 is proposed.