Abstract
Let a 0, …, a k−1 be analytic functions on a domain Ω. Let F be a family of meromorphic functions f defined on Ω such that f ≠ 0 and f (k) + a k−1 f (k−1) + … + a 0 f ≠ 0 on Ω, for all f ∊ F. Then f′ / f: f ∊ F is a normal family. Furthermore, let a 0,…, a k−1 be meromorphic functions on a domain Ω. Let F be a family of meromorphic functions f on Ω such that f≠ 0, f′ ≠ 0 and f (k) + a k− f (k−1) + … + a 0 f ≠ 0 on Ω, for all f ∊ F. Then f′ / f: f ∊ F is a normal family. These two new criteria for normal families extend a recent result of Bergweiler and Langley, [1, Corollary 1.1].
Similar content being viewed by others
References
W. Bergweiler and J. K. Langley, Nonvanishing derivatives and normal families, J. Anal. Math. 91 (2003), 353–367. MR 2 037 414
F. Brüggemann, Proof of a conjecture of Frank and Langley concerning zeros of meromorphic functions and linear differential polynomials, Analysis 12 (1992), 5–30.
J. Clunie, On integral and meromorphic functions, J. London Math. Soc. 37 (1962), 17–27. MR 26 #1456
G. Frank, Eine Vermutung von Hayman über Nullstellen meromorpher Funktionen, Math. Z. 149 no.1 (1976), 29–36. MR 54 # 10601
G. Frank and S. Hellerstein, On the meromorphic solutions of nonhomogeneous linear differential equations with polynomial coefficients, Proc. London Math. Soc. (3) 53 no.3 (1986), 407–428. MR 88k:30032
G. Frank, W. Hennekemper and G. Polloczek, Über die Nullstellen meromorpher Funktionen und deren Ableitungen, Math. Ann. 225 no.2 (1977), 145–154. MR 55 #3257
W. K. Hayman, Picard values of meromorphic functions and their derivatives, Ann. of Math. (2) 70 (1959), 9–42. MR 22 #1675
J. K. Langley, The Tsuji characteristic and zeros of linear differential polynomials, Analysis 9 no.3 (1989), 269–282. MR 90k:30056
J. K. Langley, An application of the Tsuji characteristic, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 38 no.2 (1991), 299–318. MR 92m:30060
J. K. Langley, Proof of a conjecture of Hayman concerning f and f″, J. London Math. Soc. (2) 48 no.3 (1993), 500–514. MR 94k:30075
X. Pang, Shared values and normal families, Analysis (Munich) 22 no.2 (2002), 175–182. MR 2003h:30043
X. and L. Zalcman, Normal families and shared values, Bull. London Math. Soc. 32 no.3 (2000), 325–331. MR 2001e:30059
G. Pólya, Über die Nullstellen sukzessiver Derivierten, Math. Z. 12 (1922), 36–60.
L. A. Rubel, Four counterexamples to Bloch’s principle, Proc. Amer. Math. Soc. 98 no.2 (1986), 257–260. MR 87i:30064
W. Schwick, Normality criteria for families of meromorphic functions, J. Analyse Math. 52 (1989), 241–289. MR 90k:30061
N. Steinmetz, On the zeros of (f (p) + a p−1 f (p−1) + … + a 0 f) f, Analysis 7 no.3–4 (1987), 375–389. MR 89e
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Clifford, E.F. Two New Criteria for Normal Families. Comput. Methods Funct. Theory 5, 65–76 (2005). https://doi.org/10.1007/BF03321086
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03321086