对数核Toader平均的Schur凸性和Schur几何凸性

Schur-Convexity and Schur-Geometric Concavity of Toader’s Mean with Logrithmic Kernel

为了研究对数核Toader平均Lr（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≥1/2时,Lr（a,b）在R（＋＋）2上是Schur凹函数;当r≥0时,Lr（a,b）在R（＋＋）2上是Schur几何凸函数;当r≤0时,Lr（a,b）在R（＋＋）2上是Schur几何凹函数,最后,依据Lr（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 Lr（a, b） with logrithmic kernel on R^2＋＋, using the majorization theory we obtain that Lr（a, b） is Schur-concave function on R^2＋＋ when r 〈 1 - 5, is Schur-geometric convex function on R2＋ when r 〉 0, and is Schur- geometric concave function on R^2＋＋ when r ≤ 0. Finally, new inequalities are established on the base of the Schur-convexity and Schur-geometric concavity of Lr （a, b）.