Abstract
We prove a sharp Schwarz-type lemma for meromorphic functions with spherical derivative uniformly bounded away from zero. As a consequence we deduce an improved quantitative version of a recent normality criterion due to Grahl & Nevo and Steinmetz, which is asymptotically best possibe. Based on a well--known symmetry result of Gidas, Ni & Nirenberg for nonlinear elliptic PDEs, we relate our Schwarz-type lemma to an associated nonlinear dual boundary extremal problem. As an application we obtain
Results & Lemmas (11)
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Theorem 1.1
Theorem 1.1 (Schwarz lemma for Fc) Let c > 0 and f ∈Fc. Then the following hold. (a) c ≤1/2. Mathematics Subject Classification (2000)…
Theorem 1.1 (Schwarz lemma for Fc) Let c > 0 and f ∈Fc. Then the following hold. (a) c ≤1/2. Mathematics Subject Classification (2000) Primary 30C55 1
Theorem 1.2
Theorem 1.2 Suppose that f ∈Fc for some c > 0. Then 1− q 1−4c2(1−|z0|2)2 2c(1−|z0|2)2 ≤f ♯(z0) ≤ 1+ q 1−4c2(1−|z0|2)2 2c(1−|z0|2)2, z0 ∈D.…
Theorem 1.2 Suppose that f ∈Fc for some c > 0. Then 1− q 1−4c2(1−|z0|2)2 2c(1−|z0|2)2 ≤f ♯(z0) ≤ 1+ q 1−4c2(1−|z0|2)2 2c(1−|z0|2)2 , z0 ∈D. (1.2) Both estimates are sharp if and only if z0 = 0.
Theorem 1.3
Theorem 1.3 Let c ∈(0,1/2). Then for any f ∈Fc, f ♯(z0) ≤ p 4+|z0|2 −|z0| 2 !2 1 c(1−|z0|2)2, z0 ∈D. (1.4) 2
Theorem 1.3 Let c ∈(0,1/2). Then for any f ∈Fc, f ♯(z0) ≤ p 4+|z0|2 −|z0| 2 !2 1 c(1−|z0|2)2 , z0 ∈D. (1.4) 2
Theorem 1.4
Theorem 1.4 Let c > 0 and f ∈M (D) be locally univalent such that f(0) = 0 and lim z→ξ | f ′(z)| 1+| f(z)|2 = c for all |ξ| = 1. Then f(z)…
Theorem 1.4 Let c > 0 and f ∈M (D) be locally univalent such that f(0) = 0 and lim z→ξ | f ′(z)| 1+| f(z)|2 = c for all |ξ| = 1. Then f(z) = ηz for some η ∈C. This answers the question raised by K¨uhnau in [11] (Remark after Satz 2’) who proved
Theorem 1.4
Theorem 1.4 under the additional condition that f is holomorphic on a neighborhood of D and f(D) ⊆D, and asked if this condition is…
Theorem 1.4 under the additional condition that f is holomorphic on a neighborhood of D and f(D) ⊆D, and asked if this condition is necessary. Now geometrically, it is natural to think of a function f ∈M (D) as a map f : (D,dD) →( ˆC,d ˆC) from the unit disk D equipped with the hyperbolic distance dD into the Riemann sphere ˆC equipped with the spherical distance d ˆC. We call such a map f : (D,dD) →( ˆC,d ˆC) length-preserving on the circle |z| = ρ < 1 if for each subarc γ of |z| = ρ the spheri
Corollary 1.5
Corollary 1.5 Let f ∈M (D) be a locally univalent function such that f(0) = 0 and 0 < ρ < 1. Suppose that f: (D,dD) →( ˆC,d ˆC) is…
Corollary 1.5 Let f ∈M (D) be a locally univalent function such that f(0) = 0 and 0 < ρ < 1. Suppose that f : (D,dD) →( ˆC,d ˆC) is length-preserving on the circle |z| = ρ. Then f(z) = ηz for some η ∈C. In fact, Theorem 1.1 shows that only c ≤1/2 is possible in Theorem 1.4. A short computa- tion (compare [11]) implies that this means that only ρ ≤ √ 2−1 is possible in Corollary 1.5. Hence there are locally univalent meromorphic maps f : (D,dD) →( ˆC,d ˆC), f(0) = 0, which are length–preserving o
Lemma 2.1
Lemma 2.1 Suppose that z0 ∈D 0 and w: D →D is a holomorphic function such that w(z0) = 0 and w′′(z0) = 0. Then |w′(z0)| ≤ p 4+|z0|2 −|z0|…
Lemma 2.1 Suppose that z0 ∈D\{0} and w : D →D is a holomorphic function such that w(z0) = 0 and w′′(z0) = 0. Then |w′(z0)| ≤ p 4+|z0|2 −|z0| 2(1−|z0|2) . Equality can hold only if w is a Blaschke product of degree 2. In the proof we will identify all the extremal functions semi–explicitly. We intentionally have excluded the case z0 = 0 in Lemma 2.1. 5
Theorem 3.1
Theorem 3.1 Let c > 0 and f ∈M (D) locally univalent. Then the following are equivalent: (a) lim |z|→1 f ♯(z) = c. (b) c ≤1/2 and f(z) =…
Theorem 3.1 Let c > 0 and f ∈M (D) locally univalent. Then the following are equivalent: (a) lim |z|→1 f ♯(z) = c. (b) c ≤1/2 and f(z) = T(ηz) with a rigid motion T of the Riemann sphere and |η| = 1± √ 1−4c2 2c . (3.1) In particular, this proves Theorem 1.4 which is merely a special case of Theorem 3.1.
Theorem 3.2
Theorem 3.2 (The Schwarz lemma for Fc and a dual boundary extremal problem) Let c > 0 and F ∈Fc. Then the following are equivalent. (a) F…
Theorem 3.2 (The Schwarz lemma for Fc and a dual boundary extremal problem) Let c > 0 and F ∈Fc. Then the following are equivalent. (a) F is extremal for one of the interior extremal problems max f ∈Fc f ♯(0) or min f ∈Fc f ♯(0). (b) F is extremal for the boundary extremal problem min f ∈Fc limsup z→ξ
Theorem 3.2
Theorem 3.2 roughly says that every f ∈Fc that maximizes/minimizes f ♯at the origin actu- ally minimizes f ♯on the entire unit circle. Now…
Theorem 3.2 roughly says that every f ∈Fc that maximizes/minimizes f ♯at the origin actu- ally minimizes f ♯on the entire unit circle. Now suppose that f ∈Fc maximizes/minimizes f ♯over the set Fc at a point z0 ̸= 0. Does f ♯have a corresponding boundary extremal property on (part of) the unit circle ? We are now in a position to relate the Schwarz lemma for the class Fc with Beurling’s well– known extension of the Riemann mapping theorem (see [1, 2, 4, 6, 3, 7]). Denote by H0(D) the set of all
Theorem 3.4
Theorem 3.4 (The Beurling–Riemann mapping theorem for the spherical metric) Suppose that c > 0 and consider the boundary value problem lim…
Theorem 3.4 (The Beurling–Riemann mapping theorem for the spherical metric) Suppose that c > 0 and consider the boundary value problem lim |z|→1 |g′(z)|−c 1+|g(z)|2 = 0 (3.6) for g ∈H0(D). 10
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