摘要
本文证明了: (1)如果{a_n}_n^N=1是非负不减序列,p>0,q>0,0≤r≤1,且p(q+r)≥q+p,则sum from n=1 to N(a_n^pA_n^q)(sum from m=n to N(a_n^(1+p/q)~r≤1·sum from n=1 to N(a_n^pA_n^q)^(1+p/q),其中A_n=sum from m=n to n (a_m).上述不等式在0≤r≤1时完全解决了H.Alzer^([4])在1996年提出的一个问题,且1是最佳常数; (2)如果{a_n}_n^N=1是非负序列,p,p≥1,r>0,r(p-1)≤2(q-1),令α=((p-1)(q+r)+p^2+1)/(p+1) β=(2p+2r+p-1)/(q+1),σ=(q+r-1)/(p+q+r)则sum from n=1 to N (a_n^p)sum from i=1 to n (a_i^qA_i^r)≤2~σsum from n=1 to N(a_n~αA_n~β)(0.2)(0.2)式改进了G.Be(?)et^([2,3])在1987年对Littlewood一个问题的结果,常数因子的3/2降为2^(3/2)=1.2598…
We prove : (1) If is a nondreasing sequenes, p,q > 0, 0 thenwhere, 1 is a optimal constant, a problem by H. Alzer[4] in 1996 is partiallysolved.(2) Ifis a nonnegative sequenca, p,q , letthen(0.2) provides a refinement to Littlewood's problem solved by G. Beimet2,3! in 1987.
出处
《数学的实践与认识》
CSCD
1998年第4期314-319,共6页
Mathematics in Practice and Theory
