The greatest possible lower type of entire functions of order ρ ∈ (0; 1) with zeros of fixed ρ-densities

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

“…If * = 0, by the known inequality Δ (Λ)/ Δ * (Λ) = * = 0, we have Δ (Λ) = 0. But as it was shown in work [26], each entire function with zero lower -density of zeroes has a zero lower -type.…”

“…Namely, it was proven that it is impossible in principle to estimate lower -type from above only in terms of lower -density. However, upper estimates for lower -type in terms of both -densities are possible and in [26] such precise result was proven, which stated in addition that the greatest possible lower -type is independent of the distribution of zeroes of an entire function on the plane. Theorem D (G.G.…”

“…Let us consider three cases 1) * = 0; 2) * = * ; 3) * ∈ (0, * ). In the first case the roots of equation (13) are numbers 1 = 0, 2 = and estimate (26)…”

“…Upper estimates for lower -type in terms of lower -densities similar to estimates in (1), (2) are absent in mathematical literature. The explanation of this fact was provided in [26]. Namely, it was proven that it is impossible in principle to estimate lower -type from above only in terms of lower -density.…”

“…Braichev, O.V. Sherstyukov [26]). For arbitrary order ∈ (0, 1) and arbitrary numbers 0 and > 0 ( ) the following identities hold true: I. II.…”