【作者】 何建军；

【导师】 叶中秋；

【作者基本信息】 江西师范大学 ， 应用数学， 2006， 硕士

【摘要】 本文研究了某类解析函数的单叶性半径问题和单叶函数的对数系数问题，并给出对数系数的应用。第一章中主要介绍了单叶函数及相关的一些预备知识。在第二章中设函数F（z）是单位圆盘D内的正规化的单叶函数。设 f（z）=1／（1+c）z^1-c[z^cF（z）]’，c=1，2，3，…。本章讨论了F（z）分别为α级星形函数，α级凸函数和α螺旋形函数时f（z）相应的单叶性半径，并进一步得到了当Re{F’（z）}＞α，z∈D时，使得Re{f’（z）}＞β，0≤β＜1成立的最大半径。其中有些结果是最佳的，并推广了一些作者的相关结果。第三章中引入了两个新的解析函数族K_n（α），KT_n（α），并对包含关系进行讨论，从而得到K_n（α）中的函数在整个单位圆中是单叶的。同时研究了函数族K_n（α），KT_n（α）的Golusin问题和偏差问题。第四章中研究了单叶函数的一个子族S_*^α，它是星形函数族的拓广。本章得到了S_*^α中函数的对数系数的准确的阶的估计，并用所得结论研究了相邻系数问题。更多还原

【Abstract】 Content: The problem of the radius of the univalence of some analytic functions and the logarithmic coefficients problem of univalent functions are investigated in this thesis, and we give the application of the logarithmic coefficients. In chapter 1, we introduce some preparating definitions concerning univalent functions. In chapter 2,let F(z)be regular and univalent function in the unit disk D = {z :| z |< 1} and let. The paper deals with the mapping properties of f(z) when F(z) is starlike of order α ,convex of order α ,α -spiral-like function and F(z)fits Re{F’(z)}>a. For example, if F(z)is starlike of order α, then the disk in whichf(z)is always starlike of order β (α ≤ β < 1) is determined. Some of the results aresharp,and we also improve the results obtained by some authors. In chapter 3, two new subclasses of analytic functions Kn(α),KTn(α) are introduced, and inclusionrelation has been discussed, and then we find that all the functions in Kn(α) are univalent in the unit disk. And we have studied the coefficient problem, distortion theorems in this chapter. In chapter 4, we study a new subclass S_*~α of univalentfunctions, which is the generalization of starlike functions. In this chapter we obtain the exact estimation for the order of the logarithmic coefficients and give the application on the adjacent coefficients problem.更多还原