## 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)

### MSC:

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

### Keywords:

entire function; uniqueness theorem; Nevanlinna theory

Zbl 0861.30032
Full Text:

### 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.
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