Neighborhoods of univalent functions

For an analytic function f(z) = z + Σ k = 2^∞ aₖzᵏ in the unit disc E conditions are established such that all functions g(z) = z + Σ k = 2^∞ bₖzᵏ ∈ N_δ (f), i.e. Σ k = 2^∞ k | aₖ - bₖ | δ, are in some class of univalent functions in E. For instance, we prove that every g ∈ N1/4(f) is starlike univalent in E if f is convex univalent.