Some Sharp Neighborhoods of Univalent Functions

Abstract

For $\delta \geqslant 0$ and $f(z) = z + {a_2}{z^2} + \cdots$ analytic in $|z| < 1$ let the $\delta$-neighborhood of $f$, ${N_\delta }(f)$, consist of those analytic functions $g(z) = z + {b_z}{z^2} + \cdots$ with $\sum \nolimits _{k = 2}^\infty {k|{a_k} - {b_k}| \leqslant \delta }$. We determine sufficient conditions guaranteeing which neighborhoods of certain classes of convex functions belong to certain classes of starlike functions. We extend some recent results of St. Ruscheweyh and R. Fournier and, at the same time, provide much simpler proofs. We also prove precisely how boundaries affect the value of $\delta$ for some general classes of functions.