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Normal families and multiple values. (English) Zbl 1183.30029

Let \(D\) be a domain in the complex plane and let \({\mathcal F}\) be a family of functions meromorphic in \(D\). The family \({\mathcal F}\) is called a normal family in \(D\) if each sequence \(\{f_{n}\} \subset {\mathcal F}\) has a subsequence \(\{f_{n_{j}}\}\) which converges spherically locally uniformly in \(D\) to either a meromorphic function or to \(\infty\). The authors give three basic results giving criteria for a family of functions in \(D\) to be a normal family.
Theorem 1. If \({\mathcal F}\) is a family of functions meromorphic in a domain \(D\), if \(k\) is a positive integer, and if \(b\) is a non-zero complex number such that for each \(f \in {\mathcal F}\), we have neither \(f\) nor \(f^{(k)}\) assumes the value \(0\), and the zeros of \(f^{(k)} - b\) have multiplicity at least \((k + 2)/k\), then \({\mathcal F}\) is a normal family in \(D\).
Theorem 2. If \({\mathcal F}\) is a family of functions meromorphic in a domain \(D\), if \(k, s, n_{1}, \dots, n_{q}\) are all positive integers such that \[ \frac {1} {s + k} + \sum_{j=1}^{q} \frac {1} {n_{j}} \;< q - 1 \;, \] if \(b_{1}, b_{2}, \dots, b_{q}\) are \(q\) distinct complex numbers, and if \[ F_{f}(z) = f^{(k)}(z) + \sum_{t=0}^{k-1} \;a_{t}(z) f^{(t)}(z) \;, \] where \(a_{0}, a_{1}, \dots, a_{k-1}\) are all functions analytic in \(D\), such that for each \(f \in {\mathcal F}\), if the poles and zeros of \(f\) have multiplicity at least \(s\) and \(k + 1\), respectively, and if the zeros of \(F_{f} - b_{1}, F_{f} - b_{2}, \dots, F_{f} - b_{q}\) have multiplicity at least \(n_{1}, n_{2}, \dots, n_{q}\), respectively, then \({\mathcal F}\) is a normal family in \(D\).
Theorem 3. If \({\mathcal F}\) is a family of functions meromorphic in a domain \(D\), if \(m, k, s, n_{1}, n_{2}, \dots, n_{q}\) are positive integers satisfying \[ \frac {(m + s) (k + 1)} {m(s + k)} + \sum_{j=1}^{q} \frac {1} {n_{j}} \;<q \;, \] if \(b_{1}, b_{2}, \dots, b_{q}\) are \(q\) distinct non-zero complex numbers, if \(F_{f}\) is as given in Theorem 2, if the poles and the zeros of \(f\) have multiplicity at least \(s\) and \(m \;(\geq k + 1)\), respectively, and if the zeros of \(F_{f} - b_{1}, F_{f} - b_{2}, \dots F_{f} - b_{q}\) have multiplicity at least \(n_{1}, n_{2}, \dots n_{q}\), respectively, then \({\mathcal F}\) is a normal family in \(D\).
Theorem 1 improves a result of Yong-Xin Gu (Ku, Yongxing) [Sci. Sin., Spec. Iss. 1, 267–274 (1979; Zbl 1171.30308)], Theorem 2 implies Theorem 1 (with \(q = 2, s = 1, b_{1} = 0, b_{2} = b, a_{0} \equiv a_{1} \equiv \dots \equiv a_{k-1} \equiv 0\)), and Theorem 3 improves results due to Yong-Xin Gu [Sci. Sin. 4, 431–445 (1978; Zbl 0385.30023)], J. Huang and G. D. Song [J. East China Norm. Univ., Nat. Sci. 1994, No. 3, 23–28 (1994; Zbl 0828.30017)], and X. H. Fan and X. C. Pang [J. East China Normal Univ. Nat. Sci. 2000, No. 2, 18–22 (2000)]. Some examples are given relating to the sharpness of the results. The proofs involve results from Nevanlinna theory.

MSC:

30D45 Normal functions of one complex variable, normal families
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory

Keywords:

normal family
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