Schlicht regions for entire and meromorphic functions
- 62 Downloads
Let f:C →C be a meromorpMc function. We study the size of the maximal disc inC, with respect to the spherical metric, in which a single-valued branch of f-1 exists. This problem is related to normality and type criteria. Best possible lower estimates of the size of such discs are obtained for entire functions and a class of meromorphic functions containing all elliptic functions. An estimate for the class of rational functions is also given which is best possible for rational functions of degree 7. For algebraic functions of given genus we obtain an estimate which is precise for genera 2 and 5 and asymptotically best possible when the genus tends to infinity.
Unable to display preview. Download preview PDF.
- P. P. Belinskii,General Properties of Quasiconformal Mappings, Nauka, Siberian division, 1974 (in Russian).Google Scholar
- W. Chauvenet,A Treatise on Plane and Spherical Trigonometry, J. B. Lippincott Co, Philadelphia, 1850.Google Scholar
- M. de Guzman,Differentiation of integrals in Rn, Lecture Notes in Math.421, Springer, New York, 1975.Google Scholar
- A. Lohwater and Ch. Pommerenke,On normal meromorphic functions, Ann. Acad. Sci. Fenn. Ser. A I Math.550 (1973), 1–12.Google Scholar
- D. Minda,Yosida functions, inLectures on Complex Analysis (Chuang Chi-Tai, ed.), Proc. Symp. Complex Anal., World Scientific, London, 1988, pp. 197–213.Google Scholar
- R. Nevaniirma,Analytic Functions, Springer, Berlin, 1970.Google Scholar
- Ch. Pommerenke,Estimates for normal meromorphic functions, Ann. Acad. Sci. Fenn. Ser. A I Math.476 (1970), 1–12.Google Scholar
- L. I. VolkovysM,Converging sequences of Riemann surfaces, Mat. Sb.23, 3 (1948), 361–382 (in Russian); Engl. transi.: Amer. Math. Soc. Transi. Ser. 2, Vol. 32.Google Scholar
- L. I. Volkovyskii,Researches on the type problem of a simply connected Riemann surface, Proc. Steklov Inst. Math., Acad. Sci. USSR34 1950 (in Russian).Google Scholar