Xiaofei ZHANG(張曉飛)

School of Mathematics and Statistics,Pingdingshan University,Pingdingshan 467000,China

E-mail:zhxfei@mail.ustc.edu.cn

Jin LU(盧金)?

Department of Mathematics,Huzhou University,Zhejiang 313000,China

E-mail:luking@zjhu.edu.cn

Xiaofei LI(李曉非)

School of Mathematics and Statistics,Pingdingshan University,Pingdingshan 467000,China

E-mail:lixiaofeide9999@163.com

Abstract In this article,the sharp growth theorem for almost starlike mappings of complex order λ is given firstly.Secondly,distortion theorem along a unit direction is also established as the application of the growth theorem.In particular,using our results can reduce to some well-known results.

Key words Distortion theorem;convex mapping;Schwarz lemma at the boundary

1 Introduction

Let C be the complex plane andThe unit disk in C is denoted by D.Let Cnbe the space of n complex variables z=(z1,···,zn)′with the Euclidean inner productand the Euclidean normwhere z,w∈Cnand the symbol′means transpose.The unit ball Bn={z∈ Cn:kzk<1}.Let X be a finite dimensional complex Banach space with norm k.k,B={x∈X:kxk<1}be the unit ball in X,and X?be the dual space of X.Suppose thatis a domain.Let H(?)be the set of all holomorphic mappings from ? into X.A holomorphic mapping f:? → X is said to be biholomorphic if the inverse f?1exists and it is holomorphic on the open set f(?).A mapping f ∈ H(?)is said to be normalized if f(0)=0 andDf(0)=I,where Df(0)is the Fréchet derivative of f at the origin and I is the identity operator on X.A mapping f ∈ H(?)is called locally biholomorphic if Df(x)is nonsingular at each x∈?.For each x∈X