Growth of power series with square root gaps (Q1604969)

The main result of the paper is the following theorem. Let \(f(z)=\sum_{k=1}^\infty b_{p_k}z^{p_k}\) be an entire function whose lacunary power series satisfies the condition \(p_{k+1}-p_k\geq\alpha\sqrt{p_k}\), where \(\alpha>0\). Then there exist constants \(c_\alpha\) and \(C_\alpha\), depending only on \(\alpha\), such that if \(|f(x)|\leq e^x\) for \(x\geq 0\), then \(|a_n|\leq c_\alpha\) and \(|f(z)|\leq C_\alpha e^{|z|}\), \(z\in\mathbb C\). Moreover, the author shows that the result cannot be extended to the case where \(p_{k+1}-p_k\geq\varepsilon_k\sqrt{p_k}\) with \(\varepsilon_k\searrow 0\).