Carleman不等式的新加强

Some new reinforcement of the Carleman′s inequality

运用一些分析技巧,对有限项Carleman不等式进行非严格化,给出了无限项Carleman不等式的2个新的加强式,得到了e∑nk=1kk+1αak-∑nk=1(∏ki=1ai)1/k≥Ane∑nk=11k-∑nk=1(k+1)α/k(k!)1/k;∑∞k=1(∏ki=1ai)1/k≤e∑∞k=1kk+1αak;∑∞k=1((k+1)α∏ki=1ai)1/k≤e∑∞k=1ak.其中,α=ln1 2-1≈0.442 695…,ak>0,k=1,2,…,An=1≤mki≤nn(kk+α+11)αak.

It was improved the Carleman＇s inequality, two strengthened versions of Carleman＇s inequality were given as following： e∑k=1^n（k/k＋1）^α ak-∑k=1^n（Пi=1^k ai）^1/k≥An（e∑k=1^n 1/k-∑k=1^n（k＋1）^α/k/（k!）^1/k）; ∑k=1^∞（Пi=1^k ai）^1/k≤e∑k=1^∞（k/k＋1）^α ak; ∑k=1^∞（（k＋1）^αПi=1^k ai）^1/k≤e∑k=1^∞ ak. where,α=1/ln2-1≈0.442695…,ak〉0,k=1,2…,An=min 1≤k≤n{k^α＋1/（k＋1）^α ak}