Neighborhoods of univalent functions

Abstract:

For an analytic function $f(z) = z + \Sigma _{k = 2}^\infty {a_k}{z^k}$ in the unit disc $E$ conditions are established such that all functions $g(z) = z + \Sigma _{k = 2}^\infty {b_k}{z^k} \in {N_\delta }(f)$, i.e. $\Sigma _{k = 2}^\infty k\left | {{a_k} - {b_k}} \right | \leqslant \delta$, are in some class of univalent functions in $E$. For instance, we prove that every $g \in {N_{1/4}}(f)$ is starlike univalent in $E$ if $f$ is convex univalent.