广义三角函数与双曲函数的性质的研究

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摘要

众所周知，经典的三角函数与双曲函数不仅在数学学科中有着非常重要的作用，而且在物理学、工程技术等其它学科中都有广泛的应用。1995年，Lindqvis t

给出了带p的广义三角函数sin

p x、cos

p

x、tan

p

x以及广义双曲函数

s i n h

p x、c o s h

p

x、t a n h

p

x的定义后，吸引了很多感兴趣的学者研究。

带有一个参数的广义三角函数sin

p x、cos

p

x、tan

p

x广义双曲函数sinh

p

x、

cosh

p x、tanh

p

x及其反函数arcsin

p

x、arc cos

p

x、arctan

p

x的研究是国内外最近发

展起来的新课题。当2

p 时，这些函数即为基本初等三角函数。

本文主要介绍带有单参数p的广义三角函数sin

p x,cos

p

x,tan

p

x,以及广义

双曲函数sinh

p x，cosh

p

x，tanh

p

x的一些分析性质，并用极其简单的方法解决

了文献[6]中提出的一个公开问题。同时获得了一些单参数p的广义三角函数

sin

p x,tan

p

x以及广义双曲函数sinh

p

x的Wilker型不等式以及Huygens型不等

式，这些不等式大大推广了文献[11]的一些结果。最后利用了Hölder不等式推广了Mitrinović-Adamović不等式以及Lazarević不等式。

本论文的研究内容紧密联系国际数学界所关注的前沿和热点问题，研究成果进一步完善了广义三角函数理论，同时也为与之密切相关的特殊函数如Gauss 超几何函数提供了理论基础。

关键词：广义三角函数；广义双曲函数；不等式

Abstract

It is well known that the classical trigonometric and hyperbolic function play a ve ry important role in several mathematical branches as well as in engineering and physi

cs. The generalized trigonometric sin

p x、cos

p

x、tan

p

x and hyperbolic

functions sinh

p x、cosh

p

x、tanh

p

x depending on one parameter 1

p>were studied

by P. Lindqvist in 1995. Later on numerous authors have extended this work in various directions.

Recently the generalized trigonometric and hyperbolic functions of one parameter with their inverse functions have attracted attentions of researchers at home and abroad. For the case when2

p=, these functions coincide with elementary functions.

The main research contents and expected results of this paper are as follows:

The analytic properties for the generalized trigonometric and hyperbolic functions of one parameter will be introduced. Meanwhile, the conjecture posed by the experts in [6] will be solved by using the extremely simple method. And some classical inequalities for the generalized trigonometric and hyperbolic functions of one parameters, such as Wilker-type inequality and Huygens-type inequalities are generalized. The Mitrinović-Adamovićinequality and the Lazarević inequality of generalized trigonometric and hyperbolic functions of one parameters have been extended.

The contents of the research are closely related to the hot issues of the world’s mathematics.The research results can further perfect the theory of hypergeometric function. At the same time, the research results provide theoretical basis for studying the spacial functions which are closely related them such as the Gaussian hypergeometric function.

Key words:generalized trigonometric functions; generalized hyperbolic fun ctions; inequalities