Sharp estimates for hyperbolic metrics and covering theorems of Landau type

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