双参数广义三角函数与双曲函数的几个均值不等式

Several mean inequalities for generalized trigonometric and hyperbolic functions with two parameters

双参数广义三角函数对于解决微分方程理论中的一些问题有着重要作用。文章通过初等解析方法与均值理论,研究了带有两个参数的广义三角函数与双曲函数的均值不等式性质,获得了广义正切函数与广义双曲正切函数的算术平均、调和平均以及几何平均的均值不等式;同时得到了广义三角函数的Cusa-Huygens型均值不等式。所得结果推广和改进了原有不等式性质。

Generalized trigonometric functions with two parameters play an important role in solving some problems in the theory of differential equations.The properties of mean inequalities for generalized trigonometric and hyperbolic functions with two parameters are explored by using elementary and analytic methods and the theory of means.Several new mean inequalities for generalized tangent functions and generalized hyperbolic tangent functions are obtained,such as arithmetic mean,harmonic mean and geometric mean inequalities.At the same time,Cusa-Huygens type mean inequalities for generalized trigonometric functions are obtained.The results expand and improve the properties of the original inequalities.