Treffer: on entire functions of slow growth: On entire functions of slow growth

Let \(f(z)=\sum^{\infty}_{0}a_ nz^ n\) where \(z=re^{i\theta}\) be an entire function and \(M(r)=\max_{| z| =r}| f(z)|.\) When the order of f, \(\rho =0\) i.e. the function is of slow growth, the concept of logarithmic order \(\rho^*\) (lower order \(\lambda^*)\) is defined as \[ \rho^*= \limsup_{r\to \infty}(\log \log M(r)/\log \log r), \] \[ \lambda^*=\lim \inf_{r\to \infty}(\log \log M(r)/\log \log r), \] \(1\leq \lambda^*\leq \rho^*\leq \infty\). For \(1