On entire functions which share one value CM with their first derivative
This article is devoted to considering the following conjecture: Let \(f\) be a nonconstant entire function of finite noninteger iterated order \[ \rho_1(f) = \lim\sup_{r\to \infty} \log \log T(r,f)/ \log r. \] If \(f\), \(f'\) share a value \(a\in \mathbb{C}\) counting multiplicity, then for a constant \(c\in \mathbb{C} \backslash \{0\}\), \((f'-a)/(f-a)=c\). Now, two theorems related to this conjecture will be proved: (1) The conjecture is true for \(a= 0\). (2) If \(f\) is nonconstant entire, \(N(r,{1 \over f'}) = S(r,f)\) ad \(f\), \(f'\) share \(a=1\) counting multiplicity, then for a constant \(c\in \mathbb{C} \backslash \{0\}\), \((f'-1)/(f-1) =c\). Observe that the growth condition has been dropped in (2).
scientific article; zbMATH DE number 933252