The asymptotic behaviour of certain entire functions of order zero (Q1117415)

The power series \(F(s)=\sum^{\infty}_{\nu =0}e^{-a(\nu)}s^{\nu}\) with real a(\(\nu)\) and \(a''(t)\to +\infty\) for \(t\to +\infty\) possesses the asymptotic representation F(s)\(\sim f(s)\) for \(s\to \infty\) with \(| \arg s| \leq \pi -\epsilon\) and \(\epsilon >0\), where \(f(s)=e^{-a(n)}s^ n+e^{-a(n+1)}s^{n+1}\) for \(s_ n\leq | s| \leq s_{n+1}\) with \(s_ n=\exp ((a(n+1)-a(n-1))/2).\) In case of \(a'''(t)>0\) the zeros \(s=z_ m\) of F(s) have the asymptotic behaviour \(z_ m=t_ m+O(t_{m-1})\) for \(m\to \infty\), where \(t_ m=-\exp (a(m+1)-a(m))\) are the zeros of f(s).