The Radius Of The Univalence Of Some Analytic Functions And The Logarithmic Coefficients Problem

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.