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

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”.

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 |

Full Text:
DOI

### References:

[1] | L. Ahlfors,Complex Analysis, McGraw-Hill, New York, 1979. · Zbl 0395.30001 |

[2] | H. Behnke and F. Sommer,Theorie der analytischen Funktionen einer komplexen, Veränderlichen, Springer-Verlag, Berlin, 1972. · Zbl 0273.30001 |

[3] | J. Clunie,On a result of Hayman, J. London Math. Soc.42 (1967), 389–392. · Zbl 0169.40801 |

[4] | D. Drasin,Normal families and the Nevanlinna theory, Acta Math.122 (1969), 231–263. · Zbl 0176.02802 |

[5] | G. Frank,Eine Vermutung von Hayman über Nullstellen meromorpher Funktionen, Math. Z.149 (1976), 29–36. · Zbl 0321.30034 |

[6] | G. Frank and W. Hennekemper,Einige Ergebnisse in der Werteverteilung meromorpher funktionen und ihrer Ableitungen, Resultate Math.4 (1981), 29–36. · Zbl 0491.30024 |

[7] | W. K. Hayman,Picard values of meromorphic functions and their derivatives, Ann. Math. (2)70 (1959), 9–42. · Zbl 0088.28505 |

[8] | W. K. Hayman,Meromorphic Functions, Clarendon Press, Oxford, 1964. |

[9] | G. Hennekemper,Über Fragen der Werteverteilung von Differentialpolynomen meromorpher Funktionen, Dissertation Hagen, 1979. · Zbl 0445.30024 |

[10] | K. Hiong,Sur les fonctions holomorphes dont les dérivées admettent une valeur exceptionelle, Ann. Sci. École Norm. Sup. (3)72 (1955), 165–197. · Zbl 0065.06802 |

[11] | H. Milloux,Les fonctions meromorphes et leurs derivees, Actual. Sci. Ind. No. 888, Paris (1940). · JFM 66.1249.04 |

[12] | L. Yang,Normal families and differential polynomials, Scientia Sinica (Series A)26 (1983), 673–686. · Zbl 0518.30031 |

[13] | L. Zalcman,A heuristic principle in complex function theory, Am. Math. Monthly82 (1975), 813–817. · Zbl 0315.30036 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.