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On the theorem of Tumura and Clunie. (English) Zbl 0598.30044
Let f be meromorphic and non-constant in the plane and $\phi =f\sp n+a\sb{n-1}f\sp{n-1}+...+a\sb 0$, where $a\sb j$ (0$\le j\le n-1)$ are ’small’ functions, i.e. $T(r,a\sb j)=o(T(r,f))$ as $r\to +\infty$, possibly outside a set of finite linear measure. The theorem of Tumura- Clunie [see {\it J. Clunie}, J. Lond. Math. Soc. 37, 17-27 (1962; Zbl 0104.295)] says, if f is entire and $\phi \equiv be\sp g$, b small and g entire, then $\phi \equiv (f+\frac{a\sb{n-1}}{n})\sp n$. Recently {\it N. Toda} [Contemp. Math. 25, 215-219 (1983; Zbl 0533.30030)] extended it to meromorphic functions f with $$ \overline{\lim} ess\sb{r\to +\infty}\frac{2\bar N(r,f)+\bar N(r,(1/\phi))}{T(r,f)}<\frac{1}{2}. $$ This paper proves that either $$ \phi \equiv (f+\frac{a\sb{n-1}}{n})\sp n\quad or\quad T(r,f)<\bar N(r,\frac{1}{\phi})+\bar N(r,f)+S(r,f), $$ without any condition on f or $\phi$. The proof follows by {\it E. Mues} and {\it N. Steinmetz}’s idea [J. Lond. Math. Soc., II. Ser. 23, 113-122 (1981; Zbl 0466.30025)].
Reviewer: Yang Lo

30D35Distribution of values (one complex variable); Nevanlinna theory
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