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**A note on normality and shared values.**
*(English)*
Zbl 1074.30032

Let \(D\) be a domain in the complex plane. For a complex value \(b\) in the extended complex plane, two functions \(f\) and \(g\) meromorphic in \(D\) are said to share the value \(b\) if \(f(z) = b\) if and only if \(g(z) = b\). The authors prove some sufficient conditions for a family \({\mathcal F}\) of functions meromorphic in \(D\) to be a normal family in \(D\).

Theorem 1: If \(k\) is a positive integer and \({\mathcal F}\) is a family of functions meromorphic in \(D\) such that each zero of each function \(f \in {\mathcal F}\) has multiplicity at least \(k + 2\), and if, for each pair of functions \(f, g \in {\mathcal F}\), both \((i)\) \(f\) and \(g\) share the value \(0\), and \((ii)\) \(f^{(k)}\) and \(g^{(k)}\) share a fixed value \(b\), then \({\mathcal F}\) is a normal family in \(D\).

Theorem 2: If \(k\) is a positive integer and \({\mathcal F}\) is a family of functions meromorphic in \(D\) such that each zero of each function \(f \in {\mathcal F}\) has multiplicity at least \(k + 1\), each pole of each function \(f \in {\mathcal F}\) has multiplicity at least 2, and if, for each pair of functions \(f, g \in {\mathcal F}\), both \((i)\) \(f\) and \(g\) share the value 0, and \((ii')\) \(f^{(k)}\) and \(g^{(k)}\) share a fixed value \(b\), then \({\mathcal F}\) is a normal family in \(D\).

Since functions holmorphic in \(D\) lack poles, Theorem 2 applies to a family of holomorphic functions. As a corollary, it is shown that if m is a positive integer and \({\mathcal F}\) is a family of functions meromorphic in \(D\) such that for each pair of functions \(f, g \in {\mathcal F}\), both \((i)\) \(f\) and \(g\) share the value 0, and \((ii''\)) \(f^{m+1} f'\) and \(g^{m+1} g'\) share a fixed value \(b\), then \({\mathcal F}\) is a normal family in \(D\). An example is given showing that the order of the derivatives in Theorem 1 cannot be reduced. The proofs of the theorems involve a lengthy elimination of cases, and use some results from Nevanlinna theory. The case of Theorem 1 where each \(f \in {\mathcal F}\) omits the value 0 and each \(k\)-th derivative of \(f\) omits a fixed value \(b\) is due to Y. X. Gu [Sci. Sinica, Special Issue 1, 267–274 (1979)].

Theorem 1: If \(k\) is a positive integer and \({\mathcal F}\) is a family of functions meromorphic in \(D\) such that each zero of each function \(f \in {\mathcal F}\) has multiplicity at least \(k + 2\), and if, for each pair of functions \(f, g \in {\mathcal F}\), both \((i)\) \(f\) and \(g\) share the value \(0\), and \((ii)\) \(f^{(k)}\) and \(g^{(k)}\) share a fixed value \(b\), then \({\mathcal F}\) is a normal family in \(D\).

Theorem 2: If \(k\) is a positive integer and \({\mathcal F}\) is a family of functions meromorphic in \(D\) such that each zero of each function \(f \in {\mathcal F}\) has multiplicity at least \(k + 1\), each pole of each function \(f \in {\mathcal F}\) has multiplicity at least 2, and if, for each pair of functions \(f, g \in {\mathcal F}\), both \((i)\) \(f\) and \(g\) share the value 0, and \((ii')\) \(f^{(k)}\) and \(g^{(k)}\) share a fixed value \(b\), then \({\mathcal F}\) is a normal family in \(D\).

Since functions holmorphic in \(D\) lack poles, Theorem 2 applies to a family of holomorphic functions. As a corollary, it is shown that if m is a positive integer and \({\mathcal F}\) is a family of functions meromorphic in \(D\) such that for each pair of functions \(f, g \in {\mathcal F}\), both \((i)\) \(f\) and \(g\) share the value 0, and \((ii''\)) \(f^{m+1} f'\) and \(g^{m+1} g'\) share a fixed value \(b\), then \({\mathcal F}\) is a normal family in \(D\). An example is given showing that the order of the derivatives in Theorem 1 cannot be reduced. The proofs of the theorems involve a lengthy elimination of cases, and use some results from Nevanlinna theory. The case of Theorem 1 where each \(f \in {\mathcal F}\) omits the value 0 and each \(k\)-th derivative of \(f\) omits a fixed value \(b\) is due to Y. X. Gu [Sci. Sinica, Special Issue 1, 267–274 (1979)].

Reviewer: Peter Lappan (East Lansing)

### MSC:

30D45 | Normal functions of one complex variable, normal families |

30D35 | Value distribution of meromorphic functions of one complex variable, Nevanlinna theory |

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\textit{M. Fang} and \textit{L. Zalcman}, J. Aust. Math. Soc. 76, No. 1, 141--150 (2004; Zbl 1074.30032)

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