Let f be meromorphic and non-constant in the plane and , where (0 are ’small’ functions, i.e. as , possibly outside a set of finite linear measure. The theorem of Tumura- Clunie [see J. Clunie, J. Lond. Math. Soc. 37, 17-27 (1962; Zbl 0104.295)] says, if f is entire and , b small and g entire, then . Recently N. Toda [Contemp. Math. 25, 215-219 (1983; Zbl 0533.30030)] extended it to meromorphic functions f with
This paper proves that either
without any condition on f or . The proof follows by E. Mues and N. Steinmetz’s idea [J. Lond. Math. Soc., II. Ser. 23, 113-122 (1981; Zbl 0466.30025)].