加强琴生不等式的一个猜想

A Conjecture concering Sharpening Jensen’s Inequality

本文旨在用Lagranne乘子法对于r≥2,a∈R^+建立不等式(∑a_i)~r≥∑a_i+(n^r-n)(∏a_i)^(r/n),并建立关于行列式的推广结果。

In this paper, the following results are obtained: Proposition 1 Let a∈R_n^+, r≥2.Then n^(r-1) (∑ai^r)≥(∑a_i)~r≥∑a_i^r+(n^r-n)(∏ai)^(r/n), the sign of equality holding throughout if and only if all the a_iare equal. Proposition 2 Let A_i (i=1,…, n) be real, positive definite matricesof order m, and let all the λ_i>0 (i=1,…, n) .Then |∑γ_iA_i|~r≥∑γ_i^(mr)|Ai|~r +(n^(mr)-n) (∏λ_i^m |A_i|)^(r/n)for r≥2, with equality only if A_i=k_(ij)A_j, k_(ij)>0, i≠j Proposition 3 Let A_i, B_i (i=1,…, n) be real, positive definite matri-ces of order m, and let p<1, p≠0. Then (∑|A+Bi|^(p/m))~r≥(∑|A_i|^(p/m))^(r/p)+(∑|B_i|^(p/m))^(r/p) + (2~r-2)[(∑|A_i|^(p/m)) (∑|B_i|^(p/Im))]^(r/2p)for r≥2.