TY - JOUR

T1 - Sharp estimates for hyperbolic metrics and covering theorems of Landau type

AU - Al, Baernstein II

AU - Eremenko, Alexandre

AU - Fryntov, Alexander

AU - Solynin, Alexander

PY - 2005/1

Y1 - 2005/1

N2 - In this paper we prove sharp covering theorems for holomorphic functions $f$ in the unit disk $\U$.
Theorem~1 asserts that if
$|f'(0)| \geq A|f(0)|$, where $A$ is a given number larger than $4$,
then $f$ covers some annulus of the form $r<|w|<Kr$, where
$K=K(A)$ is a number depending on $A$.
The theorem is sharp; extremals are
furnished by universal covering maps of $\U$ onto the plane minus a doubly-infinite
geometric sequence with ratio $K$ along a ray through the origin. The
covering theorem is proved by solving a minimum problem for hyperbolic
metrics. The crucial step
is to prove that among all domains $\Omega$ of the form $\C\backslash(S\times
2\pi \Z)$, where $S$ is a closed subset of $\R$ which intersects every interval
of length $\log K$, the hyperbolic density $\lambda_\Omega(z)$ is smallest when
$S$ consists of all integer multiples of $\log K$, and $z = (1/2)\log K
+ i\pi$. A second covering theorem, Theorem~2 gives the precise value
for a ``real Landau constant" about c

AB - In this paper we prove sharp covering theorems for holomorphic functions $f$ in the unit disk $\U$.
Theorem~1 asserts that if
$|f'(0)| \geq A|f(0)|$, where $A$ is a given number larger than $4$,
then $f$ covers some annulus of the form $r<|w|<Kr$, where
$K=K(A)$ is a number depending on $A$.
The theorem is sharp; extremals are
furnished by universal covering maps of $\U$ onto the plane minus a doubly-infinite
geometric sequence with ratio $K$ along a ray through the origin. The
covering theorem is proved by solving a minimum problem for hyperbolic
metrics. The crucial step
is to prove that among all domains $\Omega$ of the form $\C\backslash(S\times
2\pi \Z)$, where $S$ is a closed subset of $\R$ which intersects every interval
of length $\log K$, the hyperbolic density $\lambda_\Omega(z)$ is smallest when
$S$ consists of all integer multiples of $\log K$, and $z = (1/2)\log K
+ i\pi$. A second covering theorem, Theorem~2 gives the precise value
for a ``real Landau constant" about c

M3 - Article

SP - 113

EP - 133

JO - Ann. Acad. Sci. Fenn. Math.

JF - Ann. Acad. Sci. Fenn. Math.

ER -