Abstract
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.
Results & Lemmas (7)
Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.
Theorem 1.
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…
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.
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…
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.
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…
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.
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…
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.
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…
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
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′ =…
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
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,…
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
Related Papers