Schwarzian Derivatives and Uniform Local Univalence

Quantitative estimates are obtained for the (finite) valence of functions analytic in the unit disk with Schwarzian derivative that is bounded or of slow growth. A harmonic mapping is shown to be uniformly locally univalent with respect to the hyperbolic metric if and only if it has finite Schwarzian norm, thus generalizing a result of B. Schwarz for analytic functions. A numerical bound is obtained for the Schwarzian norms of univalent harmonic mappings. §1. Finite valence. Our point of departure is a classical theorem of Nehari [14] that gives a general criterion for univalence of an analytic function in terms of its Schwarzian derivative Sf = (f /f ) − 1 2 (f /f ) . A positive continuous even function p(x) on the interval (−1, 1) is called a Nehari function if (1 − x)p(x) is nonincreasing on [0, 1) and no nontrivial solution u of the differential equation u′′ + pu = 0 has more than one zero in (−1, 1). Nehari’s theorem can be stated as follows. Theorem A. Let f be analytic and locally univalent in the unit disk D, and suppose its Schwarzian derivative satisfies |Sf(z)| ≤ 2p(|z|) , z ∈ D , (1) for some Nehari function p(x). Then f is univalent in D. As special cases the theorem includes the criteria |Sf(z)| ≤ 2(1 − |z|2)−2 and |Sf(z)| ≤ π/2 obtained earlier by Nehari [13], as well as the criterion |Sf(z)| ≤ 4(1− |z|2)−1 stated by Pokornyi [17]. The weaker inequality |Sf(z)| ≤ 2(1 + δ ) (1− |z|2)2 , z ∈ D , 1991 Mathematics Subject Classification. Primary 30C99, Secondary 31A05, 30C55.