两个著名不等式的统一证明

Unified Proof of the Two Classical Inequalities

首先证明了数b1,b2,…,bn的一个加权平均不等式链Hn(b,P)≤Pn+∑pibiPn+∑pi/bi≤Gn(b,P)≤∑pibi/(1+bi)∑pi/(1+bi)≤An(b,P),用这个结果证明了著名的王—王不等式和kyFan不等式链Hn(b,P)Hn(1-b,P)≤Gn(b,P)Gn(1-b,P)≤An(b,P)An(1-b,P).

In this paper,we established the arithmetic,geometric,and harmonic weighted means of the numbers b_1,b_2,…,b_n:H_n(b,P)≤P_n+∑p_ib_iP_n+∑p_i/b_i≤G_n(b,P)≤∑p_ib_i/(1+b_i)∑p_i/(1+b_i)≤A_n(b,P),As a consequence we deduce the classical Inequalities of Wang_Wang and ky FanH_n(b,P)H_n(1-b,P)≤G_n(b,P)G_n(1-b,P)≤A_n(b,P)A_n(1-b,P).