The normal increasing property of entire functions (Q2751147)

A function \(f\) is a complete entire function if \(f(z)= \sum^\infty_{n=0} a_nz^n\) for each \(z\) in the plane and where \(a_i\neq 0\) for \(i= 1,2,\dots\)\ . The infinite type of \(f\) is a nondecreasing positive real function \(\Omega\) for which \(\Omega(r)\uparrow+\infty\), \(\Omega(r)\geq M(r)= \max_{|z|=r}|f(z)|\) for \(r\geq r_0\), \(\exists\) a sequence \(\{r_n\}\) tending to \(\infty\) such that \(\Omega(r_n)= M(r_n)\), and there is a nonincreasing positive continuous \(\eta\) with limit zero as \(r\) goes to infinity such that \(\Omega(r+ r/\Omega(r)^{\eta(r)})\leq \Omega(r)^{1+ \eta(r)}\) and \(\Omega(r)^{\eta(r)}\to\infty\) as \(r\to\infty\). A function of infinite type is normal increasing if there exists an \(r_0\) such that when \(r> r_0\), \(\log M(r)> (\Omega(r)\log r)^{1-\varepsilon(r)}\) and \(\log M(r)< (\Omega(r)\log r)^{1+\varepsilon(r)}\) where \(\varepsilon(r)\to 0\) as \(r\to\infty\), or if \(\lim_{r\to\infty} {\log\log M(r)\over\log(\Omega(r)\log r)}= 1\). A sample result of the authors is that a complete entire function of infinite type \(\Omega\) is normal increasing if and only if \(\lim_{r\to\infty} {\log\mu(r)\over \log(\Omega(r)\log r)}= 1\), where \(\mu(r)\) is the maximum term on \(|z|= r\) for the series for \(f\).NEWLINENEWLINEFor the entire collection see [Zbl 0966.00026].