Upper estimates for entire functions of L1(R) on real line

Анотація

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.