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 L^q[0; 2π]

Let f be an entire function, f(0) = 1, F(z) = zf′ (z)/f(z), and Γ_m = $\bigcup_{j=1}^{m}\left\{z:\mathrm{arg}z={\psi }_{j}\right\},$ 0 ≤ ψ₁ < ψ₂ < ...< ψ_m < 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 ⊂ ${\mathbb{C}}$ contained in the union of disks with finite sum of radii such that

$\begin{array}{cc}\mathrm{log}\left|f\left(z\right)\right|={\left|z\right|}^{\rho }h\left(\varphi \right)+o\left({\left|z\right|}^{\rho 2}\right),& U\not\ni z={re}^{i\varphi }\to \infty .\end{array}$

In this paper, we prove that an entire function f of order ρ ∈ (0;+∞) with zeros on a finite system of rays Γ_m is a function of improved regular growth if and only if for some ρ₂ ∈ (0; ρ) and every q ∈ [1;+∞), one has

$\begin{array}{cc}{\left\{\frac{1}{2\pi }{\int }_{0}^{2\pi }{\left|\frac{F\left({re}^{i\varphi }\right)}{{r}^{\rho }}|-\rho \widetilde{h}\left(\varphi \right)\right|}^{q}d\varphi \right\}}^{1/q}=o\left({r}^{\rho 2-\rho }\right),& r\to +\infty ,\end{array}$

where $\widetilde{h}$(φ) = h(φ) − ih′ (φ)/ρ and h(φ) is the indicator of the function f.