Neuman平均关于算术和调和平均的精确不等式

Sharp Inequalities for Neuman Mean in Terms of Arithmetic and Harmonic Means

算术、几何、二次以及调和等二元平均都是不同阶的幂平均。Schwab-Borchardt平均是一类重要的二元平均,由Schwab-Borchardt复合不同阶的幂平均可派生出一些重要平均,如Serffert平均、对数平均、Yang平均等。研究Schwab-Borchardt平均及其派生平均与不同阶幂平均的凸组合或各种特殊组合的序关系,可得到一些有价值的平均值不等式。Neuman平均是由Schwab-Borchardt平均衍生出的二元平均。本文运用实分析的方法,研究了Neuman平均与算术平均和调和平均的凸组合以及特殊组合的序关系,得到两个关于Neuman平均的精确双向不等式。

The bivariate means of arithmetic,geometry,quadratic,and harmonics are power means of different orders.Schwab-Borchardt mean compositing power means of different orders can be used to derive some important bivariate means,such as Serffert mean,logarithmic mean and Yang mean,etc.Studying the order relations of the Schwab-Borchardt mean and its derived means with convex combinations or various special combinations of the different order power means can get some valuable means inequalities.The Neuman mean is the bivariate mean derived from Schwab-Borchardt mean.In this paper,the method of real analysis is used,the order relations of the Neuman mean with the convex or special combination of arithmetic and harmonic means is studied,and two optimal bidirectional inequalities for Neuman mean are found.