Tumura-Clunie's theorem for meromorphic functions (Q1357245)

The authors of the present paper give two variations of the so-called Tumura-Clunie Theorem [\textit{J. Clunie}, J. Lond. Math. Soc. 37, 17-27 (1962; Zbl 0104.29504)] as follows: Suppose \(f\) is a non-constant meromorphic function in the plane and \(F= f^n+ a_{n-1}f^{n- 1}+\cdots+ a_0\), \(n>1\), where the \(a_j\)'s are ``small'' (i.e. \(T(r, a_j)= S(r, f)\) in Nevanlinna's terminology). Then either \(T(r, f)< N_{1)}(r, f)+ 2\overline N\left(r,{1\over f}\right)+ S(r, f)\), where \(N_{1)}\) ``counts'' the simple poles, or else \(F= \left(f+ {a_{n-1}\over n}\right)^n\). In the second theorem, \(N_{1)}(r, f)\) has to be replaced by \(N_{1)}\left(r,{1\over f+{a_{n-1}\over n}}\right)\). The techniques are customory.