p∈N的Hardy不等式的加强及加强式的自动验证

The Automatic Verification of Hardy Inequality′s Improvements and its Improving Type of p∈N

对p∈N(p≥2)的Hardy不等式的加强式进行探讨,通过对权系数W(k,p)的估计,在权系数W(k,3)下建立加强式∑∞n=1(1n∑nk=1ak)3≤278∑∞n=1[1-1n2/3(1-827W(1,3))]a3n,编写程序hdiscover,实现了形如∑∞n=1(1n∑nk=1ak)p≤(pp-1)p∑∞n=1[1-1n1-1/p(1-(p-1p)pW(1,p))]apn加强式的自动验证.猜想上式对p∈N(p≥2)恒成立.

In this paper, by means of estimating the weight coefficient W（k,p）,we discuss the Hardy Inequality＇s improving type of p∈N （p≥2） and write the program of hdiscover by establishing the improving type as ∞∑n=1（1/n n∑k=1ak）3≤27/8 ∞∑n=1[1-1/n2/3（1-8/27W（1,3））]an3, under the weight coefficient W （k, 3）,and carry out the automatic verification of Inequality＇s improving type as ∞∑n=1（1/n n∑k=1ak）p≤（p/p-1）p∞∑n=1[1-1/n1-1/p（1-（p-1/p）pW（1,p））]anp. At the end of the paper, we suppose that this inequalityrs improving type should be always valid to Hardy inequalities involving p∈N（p≥2）.