On the multiple points of certain meromorphic functions

Langley, J. K.

doi:10.1090/S0002-9939-1995-1242092-4

On the multiple points of certain meromorphic functions
HTML articles powered by AMS MathViewer

by J. K. Langley PDF

Proc. Amer. Math. Soc. 123 (1995), 1787-1795 Request permission

Abstract:

We show that if f is transcendental and meromorphic in the plane and $T(r,f) = o{(\log r)^2}$, then f has infinitely many critical values. This is sharp. Further, we apply a result of Eremenko to show that if f is meromorphic of finite lower order in the plane and $N(r,1/ff'') = o(T(r,f’ /f))$, then $f(z) = \exp (az + b)$ or $f(z) = {(az + b)^{ - n}}$ with a and b constants and n a positive integer.

References

Steven B. Bank and Robert P. Kaufman, On meromorphic solutions of first-order differential equations, Comment. Math. Helv. 51 (1976), no. 3, 289–299. MR 430370, DOI 10.1007/BF02568158
Walter Bergweiler and Alexandre Eremenko, On the singularities of the inverse to a meromorphic function of finite order, Rev. Mat. Iberoamericana 11 (1995), no. 2, 355–373. MR 1344897, DOI 10.4171/RMI/176
J. Clunie, A. Erëmenko, and J. Rossi, On equilibrium points of logarithmic and Newtonian potentials, J. London Math. Soc. (2) 47 (1993), no. 2, 309–320. MR 1207951, DOI 10.1112/jlms/s2-47.2.309
Albert Edrei and Wolfgang H. J. Fuchs, Bounds for the number of deficient values of certain classes of meromorphic functions, Proc. London Math. Soc. (3) 12 (1962), 315–344. MR 138765, DOI 10.1112/plms/s3-12.1.315
A. Erëmenko, Meromorphic functions with small ramification, Indiana Univ. Math. J. 42 (1993), no. 4, 1193–1218. MR 1266090, DOI 10.1512/iumj.1993.42.42055
Alexandre Eremenko, Jim Langley, and John Rossi, On the zeros of meromorphic functions of the form $f(z)=\sum ^\infty _{k=1}a_k/(z-z_k)$, J. Anal. Math. 62 (1994), 271–286. MR 1269209, DOI 10.1007/BF02835958
Günter Frank, Eine Vermutung von Hayman über Nullstellen meromorpher Funktionen, Math. Z. 149 (1976), no. 1, 29–36 (German). MR 422615, DOI 10.1007/BF01301627
Günter Frank and Simon Hellerstein, On the meromorphic solutions of nonhomogeneous linear differential equations with polynomial coefficients, Proc. London Math. Soc. (3) 53 (1986), no. 3, 407–428. MR 868452, DOI 10.1112/plms/s3-53.3.407
Günter Frank, Wilhelm Hennekemper, and Gisela Polloczek, Über die Nullstellen meromorpher Funktionen und deren Ableitungen, Math. Ann. 225 (1977), no. 2, 145–154. MR 430250, DOI 10.1007/BF01351718
W. K. Hayman, Picard values of meromorphic functions and their derivatives, Ann. of Math. (2) 70 (1959), 9–42. MR 110807, DOI 10.2307/1969890
W. K. Hayman, Meromorphic functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964. MR 0164038
W. K. Hayman, Research problems in function theory, The Athlone Press [University of London], London, 1967. MR 0217268
Ilpo Laine, Nevanlinna theory and complex differential equations, De Gruyter Studies in Mathematics, vol. 15, Walter de Gruyter & Co., Berlin, 1993. MR 1207139, DOI 10.1515/9783110863147
J. K. Langley, Proof of a conjecture of Hayman concerning $f$ and $f''$, J. London Math. Soc. (2) 48 (1993), no. 3, 500–514. MR 1241784, DOI 10.1112/jlms/s2-48.3.500

On second order linear differential polynomials

26

J. Miles and J. Rossi, Linear combinations of logarithmic derivatives of entire functions with applications to differential equations, Pacific J. Math. 174 (1996), no. 1, 195–214. MR 1398375, DOI 10.2140/pjm.1996.174.195
Erwin Mues, Über eine Vermutung von Hayman, Math. Z. 119 (1971), 11–20 (German). MR 276471, DOI 10.1007/BF01110938
Rolf Nevanlinna, Eindeutige analytische Funktionen, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band XLVI, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1953 (German). 2te Aufl. MR 0057330, DOI 10.1007/978-3-662-06842-7
Daniel F. Shea, On the frequency of multiple values of a meromorphic function of small order, Michigan Math. J. 32 (1985), no. 1, 109–116. MR 777306, DOI 10.1307/mmj/1029003137
Norbert Steinmetz, Rational iteration, De Gruyter Studies in Mathematics, vol. 16, Walter de Gruyter & Co., Berlin, 1993. Complex analytic dynamical systems. MR 1224235, DOI 10.1515/9783110889314