Conformal mapping, convexity and total absolute curvature

Abstract

Let f f be a holomorphic and locally univalent function on the unit disk D \mathbb {D} . Let C r C_r be the circle centered at the origin of radius r r , where 0 > r > 1 0>r >1 . We will prove that the total absolute curvature of f ( C r ) f(C_r) is an increasing function of r r . Moreover, we present inequalities involving the L p \mathrm {L}^p -norm of the curvature of f ( C r ) f(C_r) . Using the hyperbolic geometry of D \mathbb {D} , we will prove an analogous monotonicity result for the hyperbolic total curvature. In the case where f f is a hyperbolically convex mapping of D \mathbb {D} into itself, we compare the hyperbolic total curvature of the curves C r C_r and f ( C r ) 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.