On a criterion of belonging to the Hardy class $H_p(\mathbb C_{+})$ up to exponential factor

A criterion of belonging to the Hardy class $H_p(\mathbb C_{+})$ up to factor $e^{ikz}$ is obtained. It deals with functions $f$ analytic in $\mathbb C_{+}$, having Blaschke zero-sets, and satisfying the condition $|f(z)|\leq \exp\{|{\mathrm{Im}}\,z|^{-1}\exp(o(|z|))\}$, $z\to\infty$, $z\in\mathbb C_{+}$.