Abstract
We extend Carathéodory’s generalization of Montel’s fundamental normality test to “wandering” exceptional functions (i.e., depending on the respective function in the family under consideration), and we give a corresponding result on shared functions. Furthermore, we prove that if we have a family of pairs (a, b) of functions meromorphic in a domain such that a and b uniformly “stay away from each other,” then the families of the functions a resp. b are normal. The proofs are based on a “simultaneous rescaling” version of Zalcman’s lemma.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
D. Bargmann, M. Bonk, A. Hinkkanen and G. Martin, Families of meromorphic functions avoiding continuous functions, Journal d’Analyse Mathématique 79 (1999), 379–387.
C. Carathéodory, Theory of Functions of a Complex Variable, Vol. II, Chelsea Publ. Co., New York, 1960.
J. Chang, M. Fang and L. Zalcman, Composite meromorphic functions and normal families, Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 139 (2009), 57–72.
J. Clunie and W. K. Hayman, The spherical derivative of integral and meromorphic functions, Commentarii Mathematici Helvetici 40 (1966), 117–148.
J. Grahl and S. Nevo, On a result of Singh and Singh concerning shared values and normality, Complex Variables. Theory and Application 55 (2010), 347–356.
W. K. Hayman and L. A. Rubel, Unavoidable systems of functions, Mathematical Proceedings of the Cambridge Philosophical Society 117 (1995), 345–351.
P. Lappan, Avoidance criteria for normal families and normal functions, in Progress in Analysis, Vol. I, II (Berlin, 2001), World Scientific, River Edge, NJ, 2003, pp. 221–228.
P. Lappan, Special classes of normal families, Computational Methods and Function Theory 8 (2008), 133–142.
P. Lappan, A continuous function that no meromorphic function can avoid, Annales Academiae Scientiarium Fennicae. Mathematica 34 (2009), 173–177.
S. Nevo, Applications of Zalcman’s lemma to Q m-normal families, Analysis. International Mathematical Journal of Analysis and its Applications 21 (2001), 289–325.
L. A. Rubel and C.-C. Yang, Interpolation and unavoidable families of meromorphic functions, The Michigan Mathematical Journal 20 (1973), 289–296.
J. Schiff, Normal Families, Springer, New York, 1993.
A. P. Singh and A. Singh, Sharing values and normality of meromorphic functions, Complex Variables. Theory and Application 49 (2004), 417–425.
Z. Slodkowski, Holomorphic motions and polynomial hulls, Proceedings of the American Mathematical Society 111 (1991), 347–355.
L. Zalcman, A heuristic principle in complex function theory, The Amererican Mathematical Monthly 82 (1975), 813–817.
Author information
Authors and Affiliations
Corresponding author
Additional information
Research of Shahar Nevo was supported by the Israel Science Foundation, Grant No. 395/07.
Rights and permissions
About this article
Cite this article
Grahl, J., Nevo, S. Exceptional functions wandering on the sphere and normal families. Isr. J. Math. 202, 21–34 (2014). https://doi.org/10.1007/s11856-014-1054-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-014-1054-7