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

Al-Khaladi, Amer H. H.

Result. Math. 47, No. 1-2, 1-5 (2005).

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)

MSC:

30D35

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

Keywords:

entire function; uniqueness theorem; Nevanlinna theory

Citations:

Zbl 0861.30032

References:

[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.