Normality criteria for families of meromorphic functions. (English) Zbl 0667.30028

Schwick, Wilhelm

J. Anal. Math. 52, 241-289 (1989).

This paper presents the state-of-art technology of using Nevanlinna’s calculus to obtain normal-family criteria. It starts from the beginning, even recalling Nevanlinna’s “fundamental theorems”, but is beautifully developed. The idea is to show that if F is a family of meromorphic functions on a domain G of the plane, and if say \(z=0\) is in G, then \[ (*)\quad T(r,f)=O(1)\quad for\quad all\quad f\in F\quad for\quad 0<r<r_ 0, \] for some \(r_ 0>0\). Such an estimate shows that F is normal for example, near 0. The complication is that the standard calculus yields bounds in which the right side of (*) often involves intricate error terms i.e. terms which involve f and its derivatives at 0 and the problem becomes to control these.
The author’s approach depends on three ingredients. The first is now standard, and involves making infinitesimal changes of the origin to control many of these error terms, assuming always that F is not normal near 0. The second is making full use of the “Lohwater-Piranian-Zalcman principle” that if F is not normal at 0, there exist \(\rho_ j\to 0\), \(z_ j\to 0\), a sequence \(f_ j\in F\), and a nonconstant function g(z); meromorphic in the plane, such that \(f_ j(z_ j+\rho_ jz)\to g(z),\) this makes it possible to control what would otherwise be error terms of hopeless complexity. Finally, there is an interesting use of techniques of G. Frank and G./W. Hennekemper which heretofore have been applied to fixed differential polynomials.
The new criteria include: (1) if \(n\geq k+4\), \(a\neq 0\) and \(f^{(k)}(z)+P(z,f)-af^ n\) omits a fixed value b for all (f\(\in F)\), where P is a fixed linear differential polynomial with analytic coefficients involving derivatives of order \(\leq k-1\) in f; (2) if \((f^ n)^{(k)}\neq 1\) in G and either \(n\geq k+1\) with F an analytic family or \(n\geq k+3\) with F meromorphic.
The final two criteria are of a different nature. The families considered now satisfy (1) \(f^{(k)}(z)\neq 0\) for all integers k or (2) \(ff^{(k)}\neq 0\) for a fixed \(k\geq 2\) (if F is an analytic family) or \(k\geq 3\) (meromorphic case). The conclusion now is that the logarithmic derivatives of functions in F form a normal family.
Note: In the statement of Theorem 3.2, “meromorphic” should be replaced by “analytic”.

Reviewer: D.Drasin

MSC:

30D45

Normal functions of one complex variable, normal families

Keywords:

differential polynomials; normal family

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