On the spherical derivative of smoothly growing meromorphic functions with a Nevanlinna deficient value. (English) Zbl 0624.30036
The behaviour of the spherical derivative of meromorphic function with a Nevanlinna deficient value is studied. One of the results is the following Theorem 1. Let f be a transcendental meromorphic function satisfying conditions
\[
\lim_{r\to \infty}(T(2r,f)/T(r,f)) = 1
\]
and \(\delta (\infty,f)>0\). Then
\[
\overline{\lim}_{r\to \infty}(r \sup \{| f'(z)| /(1+| f(z)|^ 2)|: | z| =r\}(T(r,f))^{-1}=\infty
\]
and
\[
\overline{\lim}_{r\to \infty}n(r,0,f)\ell n(r \sup \{| f'(z)| /(1+| f(z)|^ 2): | z| =r\})(T(r,f))^{-1} \geq \delta(\infty,f).
\]
Reviewer: P.Z.Agranovich
MSC:
30D35 | Value distribution of meromorphic functions of one complex variable, Nevanlinna theory |
30D30 | Meromorphic functions of one complex variable (general theory) |