Sharp estimates for hyperbolic metrics and covering theorems of Landau type

In this paper we prove sharp covering theorems for nonconstant holomorphic functions f in the unit disk U. Theorem 1 asserts that if |f ′(0)| ≥ 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 from 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 Ω of the form C\(S × 2πZ), where S is a closed subset of R which intersects every interval of length logK, the hyperbolic density λΩ(z) is smallest when S consists of all integer multiples of logK, and z = (1/2) logK + iπ. A second covering theorem, Theorem 2, gives the precise value for a “real Landau constant” about covering intervals on the real axis whe f(0) is real. The covering and minimum problems occupy §2-§7 of the paper. In §8-§11 we study some properties of the function K(A). ∗Supported by NSF grant DMS-9801282. †Supported by NSF grants DMS-0100512, DMS-0244547 and by the Humboldt Foundation. ‡Partially supported by NSF grant DMS-0244547.