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Abstract. Let f be a holomorphic and locally univalent function on the unit disk 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 L p -norm of the curvature of f (C r ). Using the hyperbolic geometry of D, we will prove an analogous monotonicity result for the hyperbolic total curvature. In the case where f is a hyperbolically convex mapping of 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.

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