一族单叶函数的邻域

Neighbourhoods of a Class of Univalent functions

设■,其中-1≤B<A≤1,K=1,2,….设■对任意δ>0,称集?为f(x)的σ-k邻域.本文确定了I_t(A,B)中函数的σ-k邻域,一些结论是最佳的.

LetI_k(A,B)={f(z)∈?:f(z)=z+ sum from n=1 to ∞ a_nz^(nk+1),f'(z)<(1+Az)/(1+Bz)}where-1≤B<A≤1 and k=1,2,…. Let f(z)=z+ sum from n=1 to ∞ a_nz^(nk+1) belong to I_k(A,B). For δ≥0,the set {g(z) ∈ x:g(z) = z + sum from n=1 to ∞ b_nz^(nk+1), sum from n=1 to ∞(nnk+1)|a_n—b_n|≤δ} is called δ-k neighbourhood of f (z). In this paper, the δ-k neighbourhoods of functions in I_k(A,B) are characterised. Some results are best possible for some functions in I_k (A, B).