On entire functions which share one small function CM with their \(k\)th derivative. (English) Zbl 1074.30027

The author proves the following result: Let \(f\) be a non-constant entire function such that its Nevanlinna’s characteristic functions satisfy \(\overline {N} (r,\frac{1}{f^{(k)}})=S(r,f)\). If \(f\) and its \(k\)-th derivative \(f^{(k)}\) share a small meromorphic function \(a(\not\equiv 0,\infty)\) CM (counting multiplicity), then \[ f-a=(1-P_{k-1}/a)(f^{(k)}-a), \] where \(P_{k-1}\) is a polynomial of degree at most \(k-1\) such that \(1-P_{k-1}/a=e^\beta\) for an entire function \(\beta\). If \(k=1\) and if \(a\) is constant, this result is due to R. Brück [Result. Math. 30, 21–24 (1996; Zbl 0861.30032)].
Reviewer: Pei-Chu Hu (Jinan)


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


Zbl 0861.30032
Full Text: DOI


[1] AL-Khaladi, A.H.H., On entire functions which share one small function CM with their first derivative, Kodai Math.J.(to appear). · Zbl 1070.30012
[2] Brück, R., On entire functions which share one value CM with their first derivative, Results in Math., 30(1996), 21–24. · Zbl 0861.30032
[3] Hayman, W. K., Meromorphic functions, Clarendon Press, Oxford, 1964. · Zbl 0115.06203
[4] Nevanlinna, R., Le théorème de Picard-Borel et la théorie des functions méromrphes, Gauthiers-Villars, Paris, 1929.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.