Bohr Radius For The Subfamilies Of Holomorphic Functions

This dissertation mainly studies the generalizations of the Bohr radius of the unit disk and the higher dimensional extensions of the Bohr theorem.Firstly,by introducing hyperbolic metric,the Bohr theorem for shifted disks is extended to a more general case.Secondly,considering a class of positive real part functions,the corresponding Bohr radius is given by establishing its growth and covering theorem.Finally,a new norm-type higher dimensional Bohr theorem is given,and the higher dimensional Bohr radius for Carathéodory subfamily on the unit ball of complex Banach space is obtained.This thesis is divided into four chapters,and the contents of each chapter are as follows:In chapter one,we introduce the research background and main research contents.In chapter two,by introducing hyperbolic metric,the Taylor coefficient estimates of holomorphic functions on shifted disks is given,and the bohr theorem of holomorphic functions from shifted disks to convex domains is obtained.As a conclusion,we generalize the Bohr radius results of Fournier and Ruscheweyh^[6]with geometric ideas.In chapter three,considering the subclass P_α（D） of positive real part functions,the Bohr theorem is obtained by establishing growth and covering theorems of P_α（D）,and the corresponding Bohr radius is given.In chapter four,with the help of the bounded linear functionals on the complex Banach space,a new norm-type higher dimensional Bohr theorem is given.As an application,the higher dimensional Bohr radius for the Carathéodory subfamily on the unit ball B of the complex Banach space is obtained.Moreover,when B is the unit ball of the complex Hilbert space,it is proved that the constant 1/3 is the Bohr radius of the normalized convex mappings,which extends the Bohr radius of the convex functions in[9,13].