Toader型平均的两个最佳不等式

Two Optimal Inequalities for the Toader Type Mean

完全椭圆积分在物理、工程、函数论、拟共形分析、数论等相关领域有着重要的应用.通过研究Toader型平均与调和平均和正弦平均(双曲正切平均)二次凸组合的序关系,得到了Toader型平均关于调和平均和正弦平均(双曲正切平均)的上下确界.作为应用,给出了第二类完全椭圆积分关于正弦和双曲正切函数的两个精确不等式.

The complete elliptic integrals have many important applications in physics,engineering,geometric function theory,quasi-conformal analysis,theory of mean values,number theory and other related fields.By studying the order relation between Toader type mean and quadratic convex combination of harmonic mean and sine mean(hyperbolic tangent mean),we obtain that Toader type average is related to harmonic mean and sine mean(hyperbolic tangent mean).As applications,two exact inequalities for the second kind of complete elliptic integral with respect to sieand hyperbolic tangent functions are given.