Sufficient conditions for the improved regular growth of entire functions in terms of their averaging

Abstract

Let f be an entire function of order ρ∈(0,+∞) with zeros on a finite system of rays {z:argz=ψj}, j∈{1,…,m}, 0≤ψ1<ψ2<…<ψm<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 ρ1∈(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 finite sum of radii and such that log|f(z)|=|z|ρh(φ)+o(|z|ρ1), U∌z=reiφ→∞), then for some ρ3∈(0,ρ) the relation ∫r1log|f(teiφ)|tdt=rρρh(φ)+o(rρ3),r→+∞, holds uniformly in φ∈[0,2π]. In the present paper, using the Fourier coefficients method, we establish the converse statement, that is, if for some ρ3∈(0,ρ) the last asymptotic relation holds uniformly in φ∈[0,2π], then f is a function of improved regular growth. It complements similar results on functions of completely regular growth due to B. Levin, A. Grishin, A. Kondratyuk, Ya. Vasyl'kiv and Yu. Lapenko.