Alzer不等式的进一步推广

On further generalization of an inequality of H.Alzer

设 {an} ∞n =1为严格单调上升的正数列 ,给出若干条件使得下面的不等式对任意的正实数r成立 :anan+ 1<1n∑ni =1ari 1n + 1 ∑n+1i =1ari1/r,n ≥ 1 .这个结果推广了目前文献中已有的相关结果 .特别是 。

Let {a -n} +∞ - {n=1 } be a strictly increasing sequence.This paper presented several sets of conditions under any of which the inequality a -na - {n+1 }<1n∑ni=1a +r -i1n+1∑n+1i=1a +r -i + {1/r },n≥1 holds for any positive number r.These results generalize some relative ones appearing in the literature,including the Alzer′s inequality thus obtained by letting a -i=i in the above inequality.