Several Estimations Of Close-to-starlike Mappings And Spirallike Mappings

In this thesis, we study some classes of holomorphic mappings in several complex variables, and obtain the estimations of some order item coefficients of homogeneous expansion for starlike mappings, close-to-starlike mappings and spirallike mappings relative to A.The whole thesis consists of four chapters. In the first chapter, we introduce some definitions, notations and the main results of this thesis briefly. In chapter 2, on the unit ball B_n in C~n, we attain the better upper bound of second order item coefficients of homogeneous expansion for normalized starlike mappings; With the properties of Loewner chains, the upper bound of second order item coefficients of homogeneous expansion for close-to-starlike mappings are obtained; In chapter 3, we get the upper bound of second order item coefficients of homogeneous expansion for spirallike mappings relative to A via the characters of spirallike mappings relative to A on the unit ball B_n in C~n. In the last chapter, on the unit ball in a complex Banach space, we give the upper bound of second order item coefficients of homogeneous expansion for close-to-starlike mappings; Moreover, By the conception of zero of order k, we generalize the upper bound of the second order item coefficients to the upper bound from k+1 to 2k order item coefficients of homogeneous expansion for f, where f is spirallike relative to A(x = 0 is a zero of order k+1 of f(x)-x).