三元二次对称平均对幂平均的分隔及应用

The Separation of Power Means by Elementary Symmetric Mean and its Applications

Ｐ２（ｘ）是三元二次初等对称平均，Ｍｓ（ｘ）是ｓ次幂平均。本文求出了使不等式Ｍｐ（ｘ）≤Ｐ２（ｘ）≤Ｍｑ（ｘ）成立的最大ｐ和最ｑ。把这个结果用于任意ＡＢＣ，我们得到Ｍｋ（ｍａ）≥３ｒ，ｋ≥（ｌｎ３）／（ｌｎ２—ｌｎ３）；Ｍｔ（ｍａ／ａ）≥／２≥Ｍ－ｔ（ｍ／ａ），ｔ＞（ｌｎ９—ｌｎ４）／ｌｎ３；Ｍｋ（ｈａ／ｒａ）≥１，ｋ≥（ｌｎ３—ｌｎ２）／ｌｎ２；Ｍｋ（ｒａ／ｍａ）≥１，ｋ≥（ｌｎ３—ｌｎ２）／ｌｎ２，其中ａ，ｒ，ｒａ，ｈａ，ｍａ分别是边长，内切国与旁切国的半径，高以及中线。

The second elementary symmetric mean and power mean of three positive numbers x,y,zare defined byP2Mrespectively. We prove that the second elementary symmetric mean separetes two nearly powermeans in the waywhere t= (ln9-ln4)/ln3= 0. 738.. Some applications on the elements of a triangle are Mk (ma, mb,mc ) ≥3r,where K≥(ln3)/(ln2-ln3) =-2. 70...;Mt (ma/a,mb/b,mc/c)≥where t≥(ln9 -ln4 )/ln3 = 0. 738 ..;where k≥(ln3 -ln2) /In2= 0. 584..