On univalent functions convex in one direction

Abstract:

Let $f(z) = z + \Sigma _2^\infty {a_k}{z^k}$ be analytic and univalent in the unit disk $E:|z| < 1$ and map the disk onto a domain which is convex in the direction of the imaginary axis. We show by example that for $\sqrt 2 - 1 < r < 1$, the function $f(z)$ need not map the disk $|z| < r$ onto a domain convex in the direction of the imaginary axis. We also find the largest domain contained in $f(E)$ for every normalized $f(z)$ that maps E onto a domain convex in the direction of the imaginary axis.