Schwarzian Derivatives and Uniform Local Univalence

Theorem 1. Let f be analytic and locally univalent in the unit disk D, and suppose its Schwarzian derivative satisfies |Sf(z)| ≤C , z ∈D , for some constant C > π2/2 . Then |α −β| ≥ p 2/C π for any pair of points α, β ∈D where f(α) = f(β). Consequently, f has finite valence and assumes any given value at most  1 + √ 2C π

Lemma 1. Suppose that u = u(z) is a solution of the differential equation u′′ + ψu = 0 for some function ψ analytic in D. Let z = z(s) , s ∈(0, b) , be an arclength parametrization of a line segment in D, and suppose that v(z) = |u(z(s))| > 0 for s in the interval (0, b). Then v′′(s) + |ψ(z(s))| v(s) ≥0 , 0 < s < b .

Theorem 2. Let f be analytic and locally univalent in D, and suppose its Schwarzian derivative satisfies |Sf(z)| ≤ 2C 1 −|z|2 , z ∈D , (3) for a constant C > 2. Then f has finite valence N = N(C) ≤A C log C, where A is some absolute constant. The proof of Theorem 2 will invoke the separation result of Theorem 1. The following geometric lemma will be useful.

Lemma 2. If n points z1, z2, . . . , zn lie in an annulus ρ ≤|z| ≤ρ + d ≤1 and have the separation property |zj −zk| ≥2d for j ̸= k, then n ≤2π/d.

Theorem 3. Let f = h + g be an orientation-preserving harmonic mapping whose dilatation ω = g′/h′ is the square of an analytic function in the unit disk. Then ∥Sf∥< ∞if and only if f is uniformly locally univalent. The proof will invoke a recent result of Chuaqui and Hern´andez [6], which we state here for reference. Theorem B. Let f = h+g be an orientation-preserving harmonic mapping in the unit disk, and suppose that h is univalent and h(D) is convex. Then f is univalent in D.

Theorem 1′. Let f = h+g be a harmonic mapping of the unit disk with conformal parameter eσ(z) = |h′(z)|+|g′(z)| ̸= 0 and dilatation g′/h′ = q2 for some meromor- phic function q, and let ef be its lift to a minimal surface with Gauss curvature K. Suppose that |Sf(z)| + e2σ(z)|K( ef(z))| ≤C , z ∈D , for some constant C > π2/2. Then |α −β| ≥ p 2/C π for any pair of points α, β ∈D where ef(α) = ef(β). Consequently, the lift ef has finite valence and meets any given point at most  1 + √

Theorem 2′. Let a harmonic mapping f = h + g be as in Theorem 1′ but satisfy the inequality |Sf(z)| + e2σ(z)|K( ef(z))| ≤ 2C 1 −|z|2 , z ∈D , for some constant C > 2. Then its lift ef has finite valence N = N(C) ≤AC log C, where A is some absolute constant. The proofs of Theorems 1′ and 2′ reduce ultimately to the same consideration of zeros of solutions to differential equations as in the proofs of Theorems 1 and 2. Here the link with differential equations and the Sturm theory comes from a result