On neighborhoods of analytic functions with positive real part

摘要

Let h(z)=z+a_2z~2 + ··· be analytic in the unit disc U on the complex plane C. For given κ<1, ζ∈C, Rζ>κ, we prove that if ζh'(z) ∈Ω(κ, ζ) for all z∈u, then R[ζh(z)/z] >κ for all z∈u, where Ω(κ,ζ) ={w∈C: |w-2k+ζ{top}-| > |Rw-κ|} is a concave domain with boundary being a parabola. Next we consider the classes of analytic functions P(α)={p: p(0) = 1, R[p(z)] > α} and P'(α) = {p:p(0) = 1, [zp(z)]' ∈ Ω(α,1)}. For q ∈ P'(α) we find a sufficient condition on δ that implies the existence of aδ-neighborhood (N_δ)~^ being contained in P(β), where β ≤ α < 1.