The Schwarzian Derivative

Outlines

The Schwarzian Derivative Rodrigo Hern´andez Facultad de Ingenier´ıa y Ciencias, Universidad Adolfo Ib´ an ˜ez

April, 2008 Puc´ on, Chile

Rodrigo Hern´ andez

The Schwarzian Derivative

Outlines

Definitions Classical Results

Definitions

1

Introduction and Definitions Schwarzian Derivative in C Fundamental Lemmas



Outlines

Definitions Classical Results

Classical Results

2

Classical Results Nehari’s Theorems Gehring and Pommerenke Theorem p-Criterion



Introduction and Definitions

Part I Introduction and Definitions




Schwarzian Derivative in C Fundamental Lemmas

Definition

Let f : Ω ⊆ C → C be a locally biholomorphic mapping, the Schwarzian derivative is defined as Sf = (f 00 /f 0 )0 − (1/2)(f 00 /f 0 )2 . This operator characterizes the M¨ obius transformations: Sf ≡ 0 if and only if f = T , where T is a M¨ obius transformation given by T (z) =

az + b , cz + d


ad − bc 6= 0.




Definition

Let f : Ω ⊆ C → C be a locally biholomorphic mapping, the Schwarzian derivative is defined as Sf = (f 00 /f 0 )0 − (1/2)(f 00 /f 0 )2 . This operator characterizes the M¨ obius transformations: Sf ≡ 0 if and only if f = T , where T is a M¨ obius transformation given by T (z) =

az + b , cz + d


ad − bc 6= 0.




Properties

If h = f ◦ g it follows that S(f ◦ g ) = (Sf ◦ g )(g 0 )2 + Sg . Therefore, if h = f ◦ T , then Sh = (Sf ◦ T )(T 0 )2 and if h = T ◦ g , then Sh = Sg .





Properties

If h = f ◦ g it follows that S(f ◦ g ) = (Sf ◦ g )(g 0 )2 + Sg . Therefore, if h = f ◦ T , then Sh = (Sf ◦ T )(T 0 )2 and if h = T ◦ g , then Sh = Sg .





ODE

For a given analytic function p, the general function f with Sf = 2p has the form u f = , v where u and v are two linearly independent solutions of the equation. u 00 + pu = 0 .





ODE

For a given analytic function p, the general function f with Sf = 2p has the form u f = , v where u and v are two linearly independent solutions of the equation. u 00 + pu = 0 .





Comparison Theorem

Lemma Let f : Ω → C be a locally biholomorphic mapping and Ω a simply connected domain in C . Then f is univalent in Ω if and only if any nontrivial solution of u 00 + pu = 0 vanishes at most once in Ω. Lemma u 00 (z) + p(z)u(z) = 0 implies that v 00 (s) + |p(z(s))|v (s) ≥ 0 where v (s) = |u(z(s))|.





Comparison Theorem

Lemma Let f : Ω → C be a locally biholomorphic mapping and Ω a simply connected domain in C . Then f is univalent in Ω if and only if any nontrivial solution of u 00 + pu = 0 vanishes at most once in Ω. Lemma u 00 (z) + p(z)u(z) = 0 implies that v 00 (s) + |p(z(s))|v (s) ≥ 0 where v (s) = |u(z(s))|.



Classical Results

Part II Classical Results



Classical Results

Nehari’s Theorems Gehring and Pommerenke Theorem p-Criterion

Nehari’s Theorems

Theorem Let Ω be a convex domain and f : Ω → C be a locally 2π 2 biholomorphic mapping such that |Sf (z)| ≤ 2 , where δ is the δ diameter of Ω. Then f is univalent in Ω.



Classical Results


Nehari Class

The Nehari Class is defined by N = {f : D → C : (1 − |z|2 )2 |Sf (z)| ≤ 2 ,

|z| < 1} .

If f ∈ N then T ◦ f ◦ σ ∈ N when T is a M¨ obius transformation and σ is an automorphism.



Classical Results


Nehari Class

The Nehari Class is defined by N = {f : D → C : (1 − |z|2 )2 |Sf (z)| ≤ 2 ,

|z| < 1} .

If f ∈ N then T ◦ f ◦ σ ∈ N when T is a M¨ obius transformation and σ is an automorphism.



Classical Results


Nehari’s Theorems

Theorem Let f be analytic in D and suppose its Schwarzian derivative satisfies 2 |Sf (z)| ≤ , |z| < 1. (1 − |z|2 )2 Then f is univalent in D.



Classical Results


Ghering and Pommerenke Theorem Theorem Let f ∈ N then f has a continuos extension to D. f is not univalent in D if and only if f = T ◦ f ◦ σ where σ is an automorphism, T is M¨ obius transformation and 1 1+z L(z) = log . 2 1−z

SL(z) =

2 . The image is a parallel strip. (1 − z 2 )2

The function L(z) is extremal for the Nehari class N . Rodrigo Hern´ andez


Classical Results



SL(z) =




Classical Results



SL(z) =




Classical Results


p-Criterion Theorem The function f will be univalent in D if |Sf (z)| ≤ 2p(|z|) , where p(x) is a function with the following properties: p(x) > 0 and continuous for −1 < x < 1; p(−x) = p(x); (1 − x 2 )2 p(x) is nonincreasing if x varies from 0 to 1; the differential equation y 00 (x) + p(x)y (x) = 0 has a solution which not vanish for −1 < x < 1.

