Meromorphic function sharing a small function with a linear differential polynomial. (English) Zbl 1389.30136

Summary: The problem of uniqueness of an entire or a meromorphic function when it shares a value or a small function with its derivative became popular among the researchers after the work of L. A. Rubel and C.-C. Yang [Complex Anal., Proc. Conf., Lexington 1976, Lect. Notes Math. 599, 101–103 (1977; Zbl 0362.30026)]. Several authors extended the problem to higher order derivatives. Since a linear differential polynomial is a natural extension of a derivative, in the paper we study the uniqueness of a meromorphic function that shares one small function CM with a linear differential polynomial, and prove the following result: Let \(f\) be a nonconstant meromorphic function and \(L\) a nonconstant linear differential polynomial generated by \(f\). Suppose that \(a=a(z)\) (\(\not\equiv 0,\infty\)) is a small function of \(f\). If \(f-a\) and \(L-a\) share \(0\) CM and \[ (k+1)\overline{N}(r,\infty; f)+\overline{N}(r,0;f')+N_{k}(r,0;f')<\lambda T(r,f')+S(r,f') \] for some real constant \(\lambda\in (0,1)\), then \(f-a=(1+{c}/{a})(L-a)\), where \(c\) is a constant and \(1+{c}/{a}\not\equiv 0\).


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


Zbl 0362.30026
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