A new normal criterion and its application. (Chinese) Zbl 0766.30029

The author first shows (Theorem 1) that if \(f\) is an entire function, \(n\geq 2\) and \(x\geq 0\) are two integers, \(a\neq 0\) such that \(f^{(k)}+af^ n\neq 0\), then \(f\equiv\)constant. This result was then used to prove that for a non-negative integer \(k\), \(n\) integer \(\geq 2\), and \(a\), \(b\) two finite complex numbers with \(a\neq 0\), the set of all functions \(f\) that are holomorphic in a domain \(D\) satisfying, in \(D\), \(f^{(k)}+af^ n\neq b\) is a normal family.
Reviewer’s note: Theorem 1 follows immediately by applying the Tumura- Clunie type of the theorem [cf. W. K. Hayman’s book “Meromorphic functions” (1964; Zbl 0115.062), p. 69]. Thus if \(f\) is entire and satisfies \(f'+af^ 2\neq b\) (for some constants \(a\neq 0,b\)), then \(f\) must be a constant not just has an order \(\leq 2\) as asserted in the lemma 5 of the paper.


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