DISTORTION THEOREMS FOR CLASSES OF g-PARAMETRIC STARLIKE MAPPINGS OF REAL ORDER IN C~n

【摘要】 In this paper,we define the class■ of g-parametric star like mappings of real order γ on the unit ball B_X in a complex Banach space X,where g is analytic and satisfies certain conditions.By establishing the distortion theorem of the Fréchet-derivative type of■ with a weak restrictive condition,we further obtain the distortion results of the Jacobi-determinant type and the Fréchet-derivative type for the corresponding classes(compared with■) defined on the unit polydisc(resp.unit ball with the arbitrary norm) in the space of n-dimensional complex variables,n≥ 2.Our results extend the classic distortion theorem of holomorphic functions from the case in one-dimensional complex space to the case in the higher dimensional complex space.The main theorems also generalize and improve some recent works.更多还原

【Abstract】 In this paper,we define the class■ of g-parametric star like mappings of real order γ on the unit ball B_X in a complex Banach space X,where g is analytic and satisfies certain conditions.By establishing the distortion theorem of the Fréchet-derivative type of■ with a weak restrictive condition,we further obtain the distortion results of the Jacobi-determinant type and the Fréchet-derivative type for the corresponding classes(compared with■) defined on the unit polydisc(resp.unit ball with the arbitrary norm) in the space of n-dimensional complex variables,n≥ 2.Our results extend the classic distortion theorem of holomorphic functions from the case in one-dimensional complex space to the case in the higher dimensional complex space.The main theorems also generalize and improve some recent works.更多还原