Conformal mapping, convexity and total absolute curvature

Abstract:

Let $f$ be a holomorphic and locally univalent function on the unit disk $\mathbb {D}$. Let $C_r$ be the circle centered at the origin of radius $r$, where $0<r <1$. We will prove that the total absolute curvature of $f(C_r)$ is an increasing function of $r$. Moreover, we present inequalities involving the $\mathrm {L}^p$-norm of the curvature of $f(C_r)$. Using the hyperbolic geometry of $\mathbb {D}$, we will prove an analogous monotonicity result for the hyperbolic total curvature. In the case where $f$ is a hyperbolically convex mapping of $\mathbb {D}$ into itself, we compare the hyperbolic total curvature of the curves $C_r$ and $f(C_r)$ and show that their ratio is a decreasing function. The last result can also be seen as a geometric version of the classical Schwarz Lemma.