Deficiencies of composite entire functions (Q1582280)

Let \(f\) and \(g\) be transcendental entire functions with \(T(r,f) =O(\exp( (\log r)^\alpha))\) and \(T(r,g)=O ((\log r)^\beta)\) where \(0<\alpha<1\), \(\beta>1\) and \(\alpha\beta>1\). Then for any complex numbers \(a\), \(\delta(a,f (g))= \delta(a,f)\) where \(\delta\) is the Nevanlinna deficiency. The theorem generalizes Lemma 3 in \textit{R. Goldstein} [Aequationes Math. 5, 75-84 (1970; Zbl 0204.88002)] who considered the case where \(g\) is a polynomial and \(T(r,f)=O((\log r)^\alpha)\), \(\alpha>1\).