平方根平均、反调和平均和Seiffert平均的最优凸组合不等式

The Optimal Convex Combination Inequality of Square-Root and Contra-harmonic Means for the Seiffert Mean

应用初等微积分知识,找到并证明了最大值α和最小值β,使得对所有的a,b>0,a≠b双向不等式αN1(a,b)+(1-α)C(a,b)<P(a,b)<βN1(a,b)+(1-β)C(a,b)成立,其中P(a,b)、N1(a,b)、C(a,b)分别定义为两个正数a,b的Seiffert平均、平方根平均、反调和平均.

In the paper,we discover and prove by using the elementary differential calculus the greatest valuesαand the least valuesβ,such that the double inequalitiesαN1(a,b)+(1-α)C(a,b)<P(a,b)<βN1(a,b)+(1-β)C(a,b)holds for all a,b>0with a≠b,Here N1(a,b),C(a,b)and P(a,b)denote the square-root,contra-harmonic and first seiffert means of two positive real numbersαandβ,respectively.