A criterion for the improved regular growth of an entire function in terms of the asymptotic behavior of its logarithmic derivative in the metric of Lq[0;2π]

Abstract

Let f be an entire function, f(0) = 1, F(z) = zf′(z)/f(z), and Γm =Smj=1{z : arg z = ψj}, 0 ≤ ψ1 < ψ2 < . . . < ψm < 2π. An entire function f is called a function of improved regular growth if for some ρ ∈ (0; +∞) and ρ2 ∈ (0; ρ), and a 2π-periodic ρ-trigonometrically convex function h(φ) ̸≡ −∞, there exists a set U ⊂ C contained in the union of disks with fnite sum of radii such that log |f(z)| = |z|ρh(φ) + o(|z|ρ2), U ̸∋ z = reiφ → ∞. In this paper, we obtain a criterion for the improved regular growth of an entire function in terms of the asymptotic behavior of its logarithmic derivative in the metric of Lq[0; 2π].