Uniform estimates of entire functions by logarithmic sums
An entire function \(f\) of exponential type is said to be of the Cartwright class if \(\int\frac{\log^+|f(t)|}{t^2+1}dt<+\infty\). For \(B>0\) let \(\mathcal A(B)\) denote the set of all even entire functions \(f\) of the Cartwright class such that the exponential type of \(f\) is \(\leq B\), \(f|_{\mathbb R}\geq1\), and \(f(0)=1\). Define \(\mathcal I(f):=\int_0^\infty\frac{\log f(t)}{t^2}dt\), \(\mathcal S(f):=\sum_{n=1}^\infty\frac{\log f(n)}{n^2}\). The main result of the paper is the following theorem. There exists a constant \(T_\ast>0\) (\(T_\ast\simeq0.4422\)) such that if \(B<2T_\ast\), \((f_{k})_{k=1}\subset\mathcal A(B)\), and \(\mathcal S(f_{k})\longrightarrow0\), then \(\mathcal I(f_{k})\longrightarrow0\). Using the above theorem, the author proves the following two important results. (1) If \(f\) is an entire function of exponential type \(<T_\ast\) and \(\sum\frac{\log^+|f(n)|}{n^2+1}<+\infty\), then \(f\) is of the Cartwright class. (2) (A generalization of a theorem by Koosis.) \(\forall_{B_0<T_\ast}\), \(\forall_{\gamma>0}\) \(\exists_{\eta_0>0}\), \(\forall_{\eta\leq\eta_0}\) \(\exists_{C_\eta>0}\): if \(f\) is an entire function of exponential type \(\leq B\leq B_0\) with \(\sum\frac{\log^+|f(n)|}{n^2+1}\leq\eta\), then \(|f(z)|\leq C_\eta\exp(B|y|+\gamma|z|)\) for any \(z\in\mathbb C\).
scientific article; zbMATH DE number 1077108
scientific article