Nonvanishing derivatives and the MacLane class 
Alastair Fletcher, Jim Langley, and Janis Meyer
Source: Illinois J. Math. Volume 53, Number 2
(2009), 379-390.
Abstract
Let k≥2 and let f be meromorphic in the unit disc Δ, such that f(z)f(k)(z)≠0 for all z∈Δ and the poles of f in Δ have bounded multiplicities. Then f has asymptotic values on a dense subset of ∂Δ.
First Page:
Show
Hide
Full-text: Access denied (no subscription
detected)
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.ijm/1266934783
Digital Object Identifier: doi:10.1214/09-AAP195
Zentralblatt MATH identifier: 05676327
Mathematical Reviews number (MathSciNet): MR2594634
References
K. F. Barth, Asymptotic values of meromorphic functions, Michigan Math. J. 13 (1966), 321--340.
Mathematical Reviews (MathSciNet): MR0199405
Digital Object Identifier: doi:10.1307/mmj/1031732783
Project Euclid: euclid.mmj/1031732783
K. F. Barth and P. J. Rippon, Exceptional values and the Maclane class $\mathcalA$, Bergman spaces and related topics in analysis, Contemp. Math. 404 (2006), 41--52.
Mathematical Reviews (MathSciNet): MR2244003
Zentralblatt MATH: 1102.30027
W. Bergweiler and J. K. Langley, Nonvanishing derivatives and normal families, J. Analyse Math. 91 (2003), 353--367.
Mathematical Reviews (MathSciNet): MR2037414
Zentralblatt MATH: 1071.30031
Digital Object Identifier: doi:10.1007/BF02788794
J. G. Clunie, On integral and meromorphic functions, J. London Math. Soc. 37 (1962), 17--27.
Mathematical Reviews (MathSciNet): MR0143906
Digital Object Identifier: doi:10.1112/jlms/s1-37.1.17
Zentralblatt MATH: 0104.29504
G. Frank, Eine Vermutung von Hayman über Nullstellen meromorpher Funktionen, Math. Zeit. 149 (1976), 29--36.
Mathematical Reviews (MathSciNet): MR0422615
Digital Object Identifier: doi:10.1007/BF01301627
G. Frank and S. Hellerstein, On the meromorphic solutions of nonhomogeneous linear differential equations with polynomial coefficients, Proc. London Math. Soc. (3) 53 (1986), 407--428.
Mathematical Reviews (MathSciNet): MR0868452
Digital Object Identifier: doi:10.1112/plms/s3-53.3.407
G. Frank, W. Hennekemper and G. Polloczek, Über die Nullstellen meromorpher Funktionen und deren Ableitungen, Math. Ann. 225 (1977), 145--154.
Mathematical Reviews (MathSciNet): MR0430250
Digital Object Identifier: doi:10.1007/BF01351718
G. Frank and J. K. Langley, Pairs of linear differential polynomials, Analysis 19 (1999), 173--194.
Mathematical Reviews (MathSciNet): MR1705364
W. K. Hayman, Picard values of meromorphic functions and their derivatives, Ann. of Math. 70 (1959), 9--42.
Mathematical Reviews (MathSciNet): MR0110807
Digital Object Identifier: doi:10.2307/1969890
JSTOR: links.jstor.org
W. K. Hayman, Meromorphic functions, Clarendon Press, Oxford, 1964.
Mathematical Reviews (MathSciNet): MR0164038
E. L. Ince, Ordinary differential equations, Dover, New York, 1956.
Mathematical Reviews (MathSciNet): MR0010757
I. Laine, Nevanlinna theory and complex differential equations, de Gruyter Studies in Math., vol. 15, Walter de Gruyter, Berlin, 1993.
Mathematical Reviews (MathSciNet): MR1207139
J. K. Langley, Proof of a conjecture of Hayman concerning $f$ and $f'' $, J. London Math. Soc. (2) 48 (1993), 500--514.
Mathematical Reviews (MathSciNet): MR1241784
Zentralblatt MATH: 0789.30020
Digital Object Identifier: doi:10.1112/jlms/s2-48.3.500
G. R. Maclane, Asymptotic values of holomorphic functions, Rice Univ. Studies 49 (1963), 83 pp.
Mathematical Reviews (MathSciNet): MR0148923
G. R. Maclane, Exceptional values of $f^(n)(z)$, asymptotic values of $f(z)$, and linearly accessible asymptotic values, Mathematical essays dedicated to A.J. Macintyre 271--288, Ohio Univ. Press, 1970.
Mathematical Reviews (MathSciNet): MR0274765
Zentralblatt MATH: 0213.35401
G. Pólya, Über die Nullstellen sukzessiver Derivierten, Math. Zeit. 12 (1922), 36--60.
Mathematical Reviews (MathSciNet): MR1544505
Digital Object Identifier: doi:10.1007/BF01482068
W. Schwick, Normality criteria for families of meromorphic functions, J. Analyse Math. 52 (1989), 241--289.
Mathematical Reviews (MathSciNet): MR0981504
Zentralblatt MATH: 0667.30028
Digital Object Identifier: doi:10.1007/BF02820480
Y. Tumura, On the extensions of Borel's theorem and Saxer--Csillag's theorem, Proc. Phys. Math. Soc. Japan (3) 19 (1937), 29--35.
L. Zalcman, A heuristic principle in complex function theory, Amer. Math. Monthly 82 (1975), 813--817.
Mathematical Reviews (MathSciNet): MR0379852
Digital Object Identifier: doi:10.2307/2319796
JSTOR: links.jstor.org
L. Zalcman, Normal families: New perspectives, Bull. Amer. Math. Soc. (N.S.) 35 (1998), 215--230.
Mathematical Reviews (MathSciNet): MR1624862
Digital Object Identifier: doi:10.1090/S0273-0979-98-00755-1
Zentralblatt MATH: 1037.30021
2013 © University of Illinois at Urbana-Champaign, Department of Mathematics
Illinois Journal of Mathematics
