Exceptional values of meromorphic functions. (English) Zbl 0716.30023
Let \(f,c_ 1,...,c_{k-2}\) (k\(\geq 3)\) be transcendental meromorphic functions on the complex plane \({\mathcal C}\). Let \(c_ k\in {\mathcal C}- \{0\}\). Let
\[
Q=c_ kf^{(k)}+\sum^{k-2}_{m=1}c_ mf^{(m)}
\]
be not a constant. The paper, generalizing an earlier work, discusses two theorems one of which gives relations between the evB (Borel exceptional values) and the other between the evP (Picard exceptional values) of f and Q under some growth conditions on \(c_ 1,...,c_{k-2}\) in terms of f. E.g. the first theorem asserts that, if (in the usual notation of Nevanlinna theory)
\[
T(r,c_ m)=O(T(r,f))\quad as\quad r\to +\infty \text{ for } m=1,...,k-2,
\]
then either f has no evB or Q has no evB for distinct zeroes except possibly for zero in \({\mathcal C}\). The discussions are extensive and are based on six lemmas. An open problem is posed whether the \(c_ k\) could also be taken as a meromorphic function like the other \(c_ m's\).
Reviewer: J.Gopala Krishna
MSC:
30D35 | Value distribution of meromorphic functions of one complex variable, Nevanlinna theory |
30D30 | Meromorphic functions of one complex variable (general theory) |