On entire functions which share one small function CM with their first derivative. (English) Zbl 1070.30012

A meromorphic function \(a\) is said to be a small function of \(f\) provided \(T(r,a)= o (T(r,f)) \) except possibly for a set \(E\) of \(r\) of finite linear measure. Functions \(f\) and \(g\) share the small function \(a\) CM (counting multiplicities) if the equations \(f(z)-a(z) =0\) and \(g(z)-a(z)=0\) have the same zeros (counting multiplicities). The author proves the following.
Theorem. Let \(f\) be a non-constant entire function satisfying \(N(r, \frac{1}{f'})=o(T(r,f)) (r\notin E)\) and let \(a\) (not identically equal to \(0, \infty)\) be an entire small function of \(f\). If \(f\) and \(f'\) share \(a\) CM, then \(f-a=(1-\frac{k}{a})(f'-a),\) where \(1-\frac{k}{a} = e^\beta, \) \(k\) is a constant and \(\beta\) is an entire function.
The case \(a(z)\equiv 1\) was considered by R. Brück (1996).


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