Bounds on spherical derivatives for maps into regions with symmetries

S. B. Agard and F. W. Gehring,Angles and quasiconformal mappings, Proc. London Math. Soc. (3)14A (1965), 1–21. C. Carathéodory,Theory of Functions of a Complex Variable, Volume Two, Chelsea, New York, 1960. D. Hejhal,Universal covering maps for variable regions, Math. Z.137 (1974), 7–20. J. A. Hempel,The Poincaré metric on the twice punctured plane and the theorems of Landau and Schottky, J. London Math. Soc. (2)20 (1979), 435–445. E. Hopf,A remark on linear elliptic differential equations of second order, Proc. Amer. Math. Soc.3 (1952), 791–793. J. A. Jenkins,On explicit bounds in Landau’s theorem II, Can. J. Math.33 (1981), 559–562. V. JØrgensen,On an inequality for the hyperbolic measure and its applications to the theory of functions, Math. Scand.4 (1956), 113–124. D. Minda,Estimates for the hyperbolic metric, Kodai Math. J.8 (1985), 249–258. D. Minda,The hyperbolic metric and Bloch constants for spherically convex regions, Complex Variables5 (1986), 127–140. D. Minda,A reflection principle for the hyperbolic metric and applications to geometric function theory, Complex Variables8 (1987), 129–144. D. Minda and M. Overholt,The minimum points of the hyperbolic metric, Complex Variables21 (1993), 265–277. Z. Nehari,Conformal Mapping, Dover, New York, 1952. M. H. Protter and H. Weinberger,Maximum Principles in Differential Equations, Prentice-Hall, Englewood Cliffs, NJ, 1967. A. Yamada,Bounded analytic functions and metrics of constant curvature on Riemann surfaces, Kodai Math. J.11 (1988), 317–324.