一类加强不等式的推广

Extension of a Sort of Strong Inequalities

本文给出了 n个正数 x1 ,x2 ,… ,xn 的如下不等式 :∏nk=1( xαk+x-αk )≥ ( Aαn( x) +A-αn ( x) ) n ,每个 xk≤ xα,∏nk=1( xαk+x-αk )≤ ( Aαn( x) +A-αn ( x) ) n ,每个 xk≥ e.其中 α>0 ,xα=[4α2 +1 +2 α]12α ,常数 e=2 .71 81 82 81 8… ,An( x) =1n∑nk=1xk.

Inequalities on several positive number x 1,x 2,…,x n as follows: ∏nk=1(x α k+x -α k)≥(A α n(x)+A -α n(x)) n, every x k≤x α , ∏nk=1(x α k+x -α k)≤(A α n(x)+A -α n(x)) n, every x k≥ e. where α>0,x α=[4α 2+1+2α] 12α ,constant e =2 718182818…, A n(x)=1n∑nk=1x k are obtained.