Coefficient Problems For Some Subclasses Of Univalent Function

Univalent function is a kind of important analytic function in complex variable function,and harmonic mapping is a natural generalization of univa-lent function.Univalent function and its related topics are one of the most important research contents of complex variable function theory.In this pa-per,we mainly studied the coefficient estimation of several subclasses of uni-valent function,including Hankel determinant and Toeplitz determinant.As a promotion,this paper also studied the coefficient estimation of a class of close-to-convex harmonic mapping.The thesis is divided into four chapters and some specific contents are as followsIn the first chapter,the research background of coefficient estimation for univalent function is introduced firstly,then some basic concepts,marks and main lemmas are given.Finally,some main research results are listedIn the second chapter,the second Hankel determinant H2(3)for three subclasses of univalent function,namely,the classes S*(?)of starlike func-tions of order ?,C(?)of convex functions of order ? and R(?)of functions whose real part of derivative is greater than ? are studied,respectively.By using the coefficient estimates of the known function class with positive real part,the upper bounds of modules on H2(3)of three given function classes are estimated.The upper bound of module on H2(3)corresponding to the function in R(?)is sharp.Furthermore,we got the sharp bounds on H2(3)corresponding to the function in S*(?)and C(?)when the coefficient a2=0.In the third chapter,we first gived the concepts of some classes of analytic functions,i.e.two subclasses of close-to-starlike functions and two subclasses of close-to-convex functions and studied the upper bounds of module on the second order Hankel determinant H2(3)of these subclasses.then on this basis,we estimated the upper bounds of module on the generalized Zalcman functional J3,4 and the third order Toeplitz determinant T3(1)and T3(2)of these subclasses respectively,and some optimal results are obtainedIn the forth chapter,the coefficient estimation is extended to harmonic mapping,and a kind of definition of close-to-convex harmonic mapping is given The Hankel determinant and Toeplitz determinant of the conjugate analytic part are studied by using the coefficient relation between the analytic part and the conjugate analytic part of this close-to-convex harmonic mapping.