The exact bounds of lower type magnitude for entire function of order $\rho\in(0,1)$ with zeros of prescribed average densities

Abstract

We provide exact two-sided estimates for lower type magnitude of entire functions of order $\rho\in(0,1)$. The zeroes of these functions have prescribed upper and lower average densities and are arbitrarily distributed in the complex plane or on a ray. We analyze the obtained results and compare them them with known facts for entire functions of usual type.