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

Ye, Yasheng

Chin. Ann. Math., Ser. A 12, Suppl., 44-49 (1991).

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.

Reviewer: C.-C.Yang (Hongkong)

MSC:

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

Keywords:

differential polynomials; Riccati equation; normal family

Citations:

Zbl 0149.030; Zbl 0115.062