Normal criteria concerning sharing values
The author proves a new criterion for normality families of meromorphic functions. Put \[ \overline{E}_l(a,h)=\{z\in D: h(z)-a=0, \nu(z)\leq l\} , \] \[ \overline{E}(a,h)=\overline{E}_\infty(a,h)=\bigcup_{l=1}^\infty \overline{E}_l(a,h), \] where \(h\) is a meromorphic function in the domain \(D\subset \mathbb{C}\), \(a\in \mathbb{C}\), \(\nu(z)\) is the multiplicity of \(z\) as zero of the equation \(h(z)-a=0\). These notations are extended to the case \(a=\infty\) in standard way. The main result can be formulated as follows. Let us have such objects : 1) \(F\) is a family of meromorphic functions in a domain \(D\subset C\), 2) \(a_1,...,a_q\) is a set of distinct elements in \(C\cup \{\infty\}\), 3) \(l_1,...,l_q\) are strongly positive integers or infinity, such that \[ \sum_{j=1}^q\left(1-\frac{1}{l_j+1}\right)>2, \] 4) \(f_0\) is an arbitrary meromorphic function in \(D\), 5) \(b_1,..., b_p\) is a set of distinct elements in \(C\cup\{\infty\}.\) If for every \(f\in F\), \[ \bigcup_{j=1}^q \overline {E}_{l_j}(a_j,f)\subset \bigcup_{j=1}^p \overline {E}(b_j,f_0) \] then \(F\) is normal. The case \(q=3\), \(l_1=l_2=l_3=\infty\), \(\overline{E}(a_j)=\emptyset\) coincides with the theorem of Montel. The case \(\overline{E}_{l_j}(a_j,f)=\emptyset\) coincides with a theorem of Bloch and Valiron. To prove this theorem the author uses the Nevanlinna theory and a lemma of Zalcman (1975).
scientific article; zbMATH DE number 1782964