Sufficient conditions for the improved regular growth of entire functions in terms of their averaging
Let f be an entire function of order ρ(0,+∞) with zeros on a finite system of rays {z: z=ψ_j}, j{1,…,m}, 0ψ_1<ψ_2<…<ψ_m<2π and h(φ) be its indicator. In 2011, the author of the article has been proved that if f is of improved regular growth (an entire function f is called a function of improved regular growth if for some ρ(0,+∞) and ρ_1(0,ρ), and a 2π-periodic ρ-trigonometrically convex function h(φ) ot-∞ there exists a set UC contained in the union of disks with finite sum of radii and such that |f(z)|=|z|^ρ h(φ)+o(|z|^ρ_1), U ot i z=re^iφ→∞), then for some ρ_3(0,ρ) the relation equation* _1^r |f(te^iφ)|t \