Metrics with conical singularities on the sphere and sharp extensions of the theorems of Landau and Schottky

An explicit formula for the generalized hyperbolic metric on the thrice-punctured sphere ℙ\{z₁, z₂, z₃} with singularities of order α_j ≤ 1 at z_j is obtained in all possible cases α₁ + α₂ + α₃ > 2. The existence and uniqueness of such a metric was proved long time ago by Picard (J Reine Angew Math 130:243-258, 1905) and Heins (Nagoya Math J 21:1-60, 1962), while explicit formulas for the cases α₁ = α₂ = 1 were given earlier by Agard (Ann Acad Sci Fenn Ser A I 413, 1968) and recently by Anderson et al. (Math Z, to appear, doi:10. 1007/s00209-009-0560-5. We also establish precise and explicit lower bounds for the generalized hyperbolic metric. This extends work of Hempel (J Lond Math Soc II Ser 20:435-445, 1979) and Minda (Complex Var 8:129-144, 1987). As applications, sharp versions of Landau- and Schottky-type theorems for meromorphic functions are obtained.