Entire functions with real Taylor coefficients
Let \(f(z)=\sum a_ nz^ n\), Im \(a_ n=0\), be an entire transcendental function and let \(\nu_ k\) be the sequence of sign changes for \(a_ k\). The author studies the question when \[ \overline{\lim}_{r\to \infty}(\ln | f(r)| /\ln M(r,f))=1. \] Here is one of the results. Denote \(\Delta (t)=t^{-1}\sum_{\nu_ k\leq t}1,\quad D=\overline{\lim}_{t\to \infty}\Delta (t).\) The sequence \(\{\nu_ k\}\) is called regularly oscillating if \[ \sup \{(\Delta (t)/\Delta (r)):\quad t\geq r\}=O(1),\quad r\to \infty. \] Theorem 1. Let \(\{\nu_ k\}\) be a regularly oscillating sequence of positive integers. If \(D=0\) and \(\{\nu_ k\}\) is the sequence of sign changes for coefficients of some entire function of finite lower order then \[ \overline{\lim}_{r\to \infty}(\ln | f(r)| /\ln M(r,f))=1. \] If \(D>0\) then there exists an entire function of finite lower order such that \(\{\nu_ k\}\) is the sequence of sign changes for its coefficients.
scientific article; zbMATH DE number 3962061