schwarzian derivative of harmonic mappings

The classical Schwarzian derivativePre-Schwarzian and Schwarzian derivatives

Main Results

Schwarzian derivative of harmonicmappings

María J. Martín

Universidad Autónoma de Madrid,

Joint work with M. Chuaqui and R. HernándezPontificia Universidad Católica & Universidad Adolfo Ibáñez, Chile

March 15, 2013

María J. Martín Schwarzian derivative of harmonic mappings


Main Results

Let f be a locally univalent (analytic) function.

The classical Schwarzian derivative

S(f ) =

(f ′′

f ′

)′− 1

2

(f ′′

f ′

)2

= P(f )z −12

P(f )2 ,

P(f ) =f ′′

f ′=

∂

∂z

(log |f ′|2

).

Classical notation (Cayley, 1880):

S(f )(z) = {f , z} .



Main Results

Hermann Amandus Schwarz, 1843–1921Define an operator S with the property that S(T ◦ f ) = S(f )for all linear fractional transformations

T (z) =az + bcz + d

, ad − bc 6= 0 ,

and all locally univalent functions f .

Let g = T ◦ f . Then,

g =af + bcf + d

orc(fg) + dg − af = b .



Main Results

c(fg) + dg − af = b .

Taking derivatives, we getc(fg)′ + dg′ − af ′ = 0

c(fg)′′ + dg′′ − af ′′ = 0c(fg)′′′ + dg′′′ − af ′′′ = 0

.

Hence, (f ′′

f ′

)′− 1

2

(f ′′

f ′

)2

≡(

g′′

g′

)′− 1

2

(g′′

g′

)2

.

S(T ◦ f ) = S(f ) .



Main Results

S(T ◦ f ) = S(f ) .

A function f has Schwarzian Sf = 2p if and only if f = u1/u2where u1 and u2 are independent solutions of

u′′ + pu = 0.

Sf = Sg ⇐⇒ g = T ◦ f .

Chain Rule

S(f ◦ g)(z) = Sf (g(z)) · g′(z)2 + Sg(z) .



Main Results

Let φ be an automorphism of D with φ(0) = z. Then,

S(f ◦ φ)(0) = S(f )(z)(1− |z|2)2 .

The Schwarzian norm

‖S(f )‖ = supz∈D|S(f )(z)|(1− |z|2)2 .

Krauss, 1932If f is univalent in the unit disk, then ‖S(f )‖ ≤ 6 .

Nehari, 1949If ‖S(f )‖ ≤ 2 , then f is univalent.



Main Results

The pre-Schwarzian norm

‖P(f )‖ = supz∈D|P(f )(z)|(1− |z|2) .

If f is univalent in the unit disk, then ‖P(f )‖ ≤ 6 .

If ‖P(f )‖ ≤ 1 , then f is univalent.

Use S to denote the family of all analytic functions f in theunit disk which are univalent in D and satisfy

f (0) = f ′(0)− 1 = 0 .



Main Results

Harmonic mappingsDefinition and properties

The function f = u + iv : D→ C is harmonic if

∆f =∂2f∂x2 +

∂2f∂y2 = 4

∂2f∂z∂z

≡ 0, z = x + iy .

The canonical decomposition of f equals h + g, where h,g ∈ H(D), and g(0) = 0.

Lewy, 1936

The harmonic mapping f = h + g is locally univalent if andonly if

Jf = |h′|2 − |g′|2 6= 0 .

We call f orientation-preserving if Jf > 0 ≡ h is locallyunivalent and |ωf | = |g′/h′| < 1.



Main Results


Use SH to denote the family of all orientation-preservingharmonic functions f = h + g that are univalent in D withthe normalizations

h(0) = g(0) = h′(0)− 1 = 0 .

SH is a normal family, but it is not compact:

fn(z) = z +n

n + 1z .

TheoremThe family

S0H = {f ∈ SH : g′(0) = 0}

is normal and compact .



Main Results


Take f ∈ SH with |ωf (0)| = |g′(0)| = |a| < 1. Consider

ϕ(w) =w − aw1− |a|2

.

Then,f0 = ϕ ◦ f ∈ S0

H .

Also, given f0 ∈ S0H and a ∈ D, the function

f = f0 + af0

belongs to SH and satisfies g′(0) = a.



Main Results


The harmonic Koebe function

K (z) =z − 1

2z2 + 16z3

(1− z)3 +

(12z2 + 1

6z3

(1− z)3

)∈ S0

H .

K = h + g, where h and g satisfies{h− g = k , k(z) = z

(1−z)2

g′/h′ = Id, h(0) = g(0) = 0 .

Clunie and Sheil-Small, 1984Let f = h + g be locally univalent. Then, f is univalent andconvex in the horizontal direction (CHD) if and only if theanalytic function h− g is univalent and CHD.



Main Results


An orientation-preserving harmonic mapping f = h + gcan be lifted (locally) to a regular surface given byisothermal parameters if and only if ωf = q2 for someanalytic function q (with |q| < 1). The minimal surface hasthe Weierstrass-Enneper representation

u = Re{∫ z

z0

h′(1 + q2)dζ}, v = Im

{∫ z

z0

h′(1− q2)dζ},

and

w = 2 Im{∫ z

z0

h′qdζ}.

The metric of the surface has the form ds = ρ|dz| whereρ = |h′|+ |g′| > 0.



Main Results


Sf = 2(σzz − σ2z ) ,

where σ = log(|h′|+ |g′|) .

In terms of the canonical representation f = h + g and thedilatation ω = q2,

Chuaqui, Duren, and Osgood, 2003

S(f ) = S(h) +2q

1 + |q|2

(q′′ − q′

h′′

h′

)− 4

(q′q

1 + |q|2

)2

.



Main Results


Properties

S(f ◦ φ) = S(f )(φ(z))φ′(z)2 + S(φ)(z) .

S(f ) is analytic if and only if f = h + ah with |a| < 1 andS(h) = S(f ) .‖S(f )‖ <∞ if and only if ‖S(h)‖ <∞ if and only if f isuniformly locally univalent.There exists a constant M such that

‖S(f )‖ = supz∈D|S(f )(z)|(1− |z|2)2 ≤ M

for all univalent harmonic mappings f with ωf = q2.



Main Results


Disadvantage

If f is an orientation-preserving harmonic mapping withωf = q2, a ∈ D, and L(z) = z + az, then F = L ◦ f hasdilatation

ωF =q2 + a1 + aq2 .



Main Results


DefinitionLet f = h + g be a sense-preserving harmonic mappingwith dilatation ω = g′/h′. We define

Sf = S(h) +ω

1− |ω|2

(h′′

h′ω′ − ω′′

)− 3

2

(ω′ ω

1− |ω|2

)2

.



Main Results


Sf = S(h) +ω

1− |ω|2

(h′′

h′ω′ − ω′′

)− 3

2

(ω′ ω

1− |ω|2

)2

????????

Approximating by Möbius transformations

Let f be LU in D. Consider

Tf (z) =az + bcz + d

, ad − bc 6= 0 ,

with Tf (0) = f (0), T ′f (0) = f ′(0), and T ′′f (0) = f ′′(0).

Ff (z) = (T−1f ◦ f )(z) =

∞∑n=0

snzn

n!= z +

13!

S(f )(0) z3 + . . . ,

S(f )(w) = S(Ff )(w) .



Main Results


Let f = h + g be sense-preserving in D. Consider

Mf = T + αT , T (z) =az + bcz + d

, |α| < 1 ,

with Mf (0) = f (0),

∂Mf

∂z(0) =

∂f∂z

(0) = h′(0),∂Mf

∂z(0) =

∂f∂z

(0) = g′(0) ,

and∂2Mf

∂z2 (0) =∂2f∂z2 (0) = h′′(0) .

Ff (z) = (M−1f ◦ f )(z) .



Main Results


Sf = S(h) +ω

1− |ω|2

(h′′

h′ω′ − ω′′

)− 3

2

(ω′ ω

1− |ω|2

)2

.

Ff (z) = z − 12!

(ω(0)ω′(0)

1− |ω(0)|2

)z2 +

12!

(h′(0)ω′(0)

h′(0)(1− |ω(0)|2)

)z2

+13!

(Sh(0) +

ω(0)

1− |ω(0)|2

(ω′(0)

h′′

h′(0)− ω′′(0)

))z3

− 13!

(h′′(0)h′(0)ω′(0)

(h′(0))2(1− |ω(0)|2)

)zz2

− 13!

(1

1− |ω(0)|2

(ω′′(0) + 2ω′(0)

h′′

h′(0)

))z3 + · · ·



Main Results


If φ is analytic and locally univalent,

S(φ) = ρzz −12ρ2

z = P(φ)z −12

P(φ)2 ,

where ρ = log(Jf ) (and P(φ) = ∂∂z (log Jf )).

The same formula holds for locally univalent harmonicfunctions:

Sf = σzz −12σ2

z = (Pf )z −12

(Pf )2 ,

where σ = log(Jf ) and

Pf =∂

∂z(log Jf ) =

h′′

h′− ω ω′

1− |ω|2.



Main Results


Properties

S(f◦φ)(z) = Sf (φ(z))φ′(z)2 + S(φ)(z) .

Sf is analytic if and only if f = h + ah with |a| < 1 andS(h) = Sf .‖S(f )‖ <∞ if and only if ‖S(h)‖ <∞ if and only if f isuniformly locally univalent.

supf∈SH

‖Sf‖ <∞ .

Take L(z) = az + bz with |a| 6= |b|. Then, S(L◦f ) ≡ Sf .



Main Results


Properties

P(f◦φ)(z) = Pf (φ(z))φ′(z) + P(φ)(z) .

Pf is analytic if and only if f = h + ah with |a| < 1 andP(h) = Pf .

supf∈SH

‖Pf‖ <∞ .

Take L(z) = az + bz with |a| 6= |b|. Then, P(L◦f ) ≡ Pf .



Main Results

Univalence and pre-SchwarzianQuasiconformal extensionsOrder of LI families

TheoremIf f = h + g is a sense-preserving harmonic mapping in theunit disk with second complex dilatation ω and

|Pf (z)|(1− |z|2) +|ω′(z)|(1− |z|2)

1− |ω(z)|2≤ 1

for all |z| < 1, then f is univalent in D. The constant 1 issharp.

The condition in the previous theorem implies that h + agis univalent for all |a| ≤ 1.



Main Results


The Jacobian of a function f of complex values equalsJf = |fz |2 − |fz |2. For sense-preserving mappings w = f (z),we have

(|fz | − |fz |) |dz| ≤ |dw | ≤ (|fz |+ |fz |) |dz| .

Df =|fz |+ |fz ||fz | − |fz |

.

A sense-preserving homeomorphism f is K -quasiconformalif f ∈W 1,2

loc and Df ≤ K(or |µf | = |fz/fz | = |ωf | ≤ k = (K − 1)/(K + 1) < 1).



Main Results


Becker and Ahlfors, 1972; 1974If

supz∈D|Pφ(z)| (1− |z|2) ≤ k < 1 ,

then φ has a K -quasiconformal extension to the wholecomplex plane C, where K = (1 + k)/(1− k).An explicit quasiconformal extension is defined by

Φ(z) =

φ̃(z), |z| ≤ 1

φ(

1z

)+ u

(1z

), |z| > 1

,

where, for z ∈ D \ {0}, u(z) = φ′(z)(1− |z|2)/z.



Main Results


Theorem 1Assume that

|Pf (z)|(1− |z|2) + |ω∗(z)| ≤ k < 1, z ∈ D .

Then, f has a continuous and injective extension f̃ to Dand

F(z) =

f̃ (z), |z| ≤ 1

f(

1z

)+ U

(1z

), |z| > 1

is a homeomorphic extension of f to the whole complexplane onto itself. The function U equals

U(z) =h′(z)(1− |z|2)

z+

g′(z)(1− |z|2)

z, z ∈ D \ {0} .



Main Results


Theorem 2

If, in addition, ‖w‖∞ < 1, then f̃ (∂D) is a quasicircle and fcan be extended to a quasiconformal map in C. Indeed,the function F is an explicit K -quasiconformal extension off whenever

k <1− ‖w‖∞1 + ‖w‖∞

.

The constant K equals

K =(1 + k) + (1− k)‖w‖∞(1− k)− (1 + k)‖w‖∞

.



Main Results


Let F be a family of locally univalent holomorphicfunctions f in the unit disk normalized by the conditionsf (0) = 1− f ′(0) = 0. If F closed under the Koebe transformdefined by

Fζ(z) =

f(ζ + z

1 + ζz

)− f (ζ)

(1− |ζ|2)f ′(ζ), ζ ∈ D,

we call F a linear invariant family. The order of F is

α(F) = supf∈F|a2(f )| =

12

supf∈F|f ′′(0)| .



Main Results


Pommerenke, 1964

Fλ = {f : D→ C : f ′ 6= 0 in D, f (0) = 0, f ′(0) = 1, ‖Sf‖ ≤ λ}

is

α(Fλ) =

√1 +

λ

2.



Main Results


Let F be a family of sense-preserving harmonic mappingsf = h + g in D, normalized with h(0) = g(0) = 0,h′(0) = 1.The family is said to be affine and linearly invariant (ALfamily) if it closed under the two operations:

Kζ(f )(z) =

f(

z + ζ

1 + ζz

)− f (ζ)

(1− |ζ|2)h′(ζ), (1)

and

Aε(f )(z) =f (z) + εf (z)

1 + εg′(0). (2)

The order of the family, given by

α(F) = supf∈F|a2(h)| =

12

supf∈F|h′′(0)| .



Main Results


The order of the family Fλ satisfies

α (Fλ) ≤√λ

2+ 1 +

12

supf∈F0

λ

|g′′(0)|2 +12

supf∈Fλ

|g′(0)|

<

√λ

2+

32

+12.

Moreover,

S =12

supf∈F0

λ

|h′′(0)| ≤√λ

2+

32.


schwarzian derivative of harmonic mappings

Documents