Asymptote of some entire exponential-type functions with zeros on spirals

The author considers the Weierstrass canonical product of the first kind $\Pi(z)$, all roots of which lie on a spiral with equation in polar coordinates $(r,\Phi):\Phi=\ln\ln r$. With certain additional conditions on the roots, the asymptote is found for the function $\ln\{e^{Az}\Pi(z)\}$ ($A$ is some constant) in the complex plane cut along the spiral $\Phi=\ln\ln r$. The result is applied to the question of the sufficient condition for the satisfaction of an inequality for exponential-type functions, used in questions of the Dirichlet-series representation of analytic functions.