Upper estimates for entire functions of $L^1(R)$ on real line

A. Il'inskii

Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 7 (2000) no. 2, pp. 172-183

Résumé

Let $\mathcal S_{\rho}$ be the set of all entire functions of order $\rho$ and normal type such that $f(x)\ge 0$ for $x\in\mathbf R$ and $f\in L^1(\mathbf R)$. We prove that: 1) if $f\in\mathcal S_{\rho}$, then $f(x)=o(|x|^{\rho-1})$, $x\to\pm\infty$, 2) for any sequence $\varepsilon_n\downarrow 0$ there exists a function $f\in\mathcal S_{\rho}$ and a real sequence $b_n\to+\infty$ such that $f(b_n)>b_n^{\rho-1-\varepsilon_n}$. We give a generalization of this result for more general growth scale.