Convex and starlike meromorphic functions

Miller, James

doi:10.1090/S0002-9939-1980-0587937-5

Convex and starlike meromorphic functions
HTML articles powered by AMS MathViewer

by James Miller

Proc. Amer. Math. Soc. 80 (1980), 607-613

DOI: https://doi.org/10.1090/S0002-9939-1980-0587937-5

PDF | Request permission

Abstract:

In this paper we study meromorphic univalent functions with a simple pole at p which map the unit disk onto the exterior of convex sets or the exterior of starlike sets. A question raised by Libera and Livingston concerning the equality of two classes of starlike meromorphic univalent functions is answered. We show that the set of points omitted by the class of convex functions is $\{ - p/(1 + {p^2})\}$. The coefficients of convex and starlike meromorphic functions are also studied.

References

J. Clunie, On meromorphic schlicht functions, J. London Math. Soc. 34 (1959), 215–216. MR 107009, DOI 10.1112/jlms/s1-34.2.215
Richard J. Libera and Albert E. Livingston, Weakly starlike meromorphic univalent functions, Trans. Amer. Math. Soc. 202 (1975), 181–191. MR 361036, DOI 10.1090/S0002-9947-1975-0361036-2
James Miller, Convex meromorphic mappings and related functions, Proc. Amer. Math. Soc. 25 (1970), 220–228. MR 259098, DOI 10.1090/S0002-9939-1970-0259098-7
John A. Pfaltzgraff and Bernard Pinchuk, A variational method for classes of meromorphic functions, J. Analyse Math. 24 (1971), 101–150. MR 281899, DOI 10.1007/BF02790372
W. C. Royster, Convex meromorphic functions, Mathematical Essays Dedicated to A. J. Macintyre, Ohio Univ. Press, Athens, Ohio, 1970, pp. 331–339. MR 0273002
T. Sheil-Small, On convex univalent functions, J. London Math. Soc. (2) 1 (1969), 483–492. MR 252625, DOI 10.1112/jlms/s2-1.1.483
T. J. Suffridge, Some remarks on convex maps of the unit disk, Duke Math. J. 37 (1970), 775–777. MR 269827