Length and Area Estimates for (Hyperbolically) Convex Conformal Mappings

Abstract

Let f be a conformal map on ${{\mathrm{\mathbb {D}}}}$. Keogh (J Lond Math Soc 29:121–123, 1954) obtained an upper bound for the length of the curve $f(|z|=r)$, $r \in (0,1)$, when f is a convex map. With the use of this upper bound, we prove a monotonicity result regarding the length of $f(|z|=r)$. A similar kind of monotonicity result is proved for the area of $f(|z|<r)$. Considering the case where $f({{\mathrm{\mathbb {D}}}}) \subset {{\mathrm{\mathbb {D}}}}$ and f is a hyperbolically convex map, we present hyperbolic analogues of the above bounds. We prove two monotonicity theorems regarding the length and area in the hyperbolic geometry of the unit disk. In particular, the ratio of the hyperbolic length of the curve $f(|z|=r)$ to the hyperbolic length of the curve $|z|=r$ is a decreasing function of r. Furthermore, we prove that the ratio of the hyperbolic area of $f(|z|<r)$ to the hyperbolic area of the disk $|z|<r$ is also a decreasing function of r.