线性核Toader平均的Schur凸性和Schur几何凸性

Schur-Convexity and Schur-Geometric Concavity of Toader's Mean with Linear Kernel

为了研究线性核Toader平均Mr（a,b）在R＋＋2上的Schur凸性和Schur几何凸性,利用控制不等式的相关理论得到结论：当r≥1时,Mr（a,b）在R＋＋2上是Schur凸函数;当r≤1时,Mr（a,b）在R＋＋2上是Schur凹函数;当r≥1/2时,Mr（a,b）在R＋＋2上是Schur几何凸函数.最后,依据Mr（a,b）的Schur凸性和Schur几何凸性建立了新的不等式.

In order to research the Schur-convexity of Toader’s mean Mr（a,b） with linear kernel on R＋＋2,using the majorization theory we obtain that Mr（a,b） is Schur-convex function on R＋＋2 when r≥1,is Schur-concave function on R＋＋2 when r≤1,and is Schur-geometric convex function on R＋＋2 when r≥（1/2）.Finally,new inequalities are established on the base of the Schur-convexity and Schur-geometric concavity of Mr（a,b）.