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]

Let f be an entire function, f(0) = 1, F(z) zf′ (z)/f(z), and Γm $$\bigcup_{j=1}^{m}\left\{z:\mathrm{arg}z={\psi }_{j}\right\},$$ 0 ≤ ψ1 < ψ2 ...< ψm 2π. An function is called a of improved regular growth if for some ρ ∈ (0;+∞) ρ2 (0; ρ), 2π-periodic ρ-trigonometrically convex h(φ) ≢ −∞, there exists set U ⊂ $${\mathbb{C}}$$ contained in the union disks with finite sum 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 order zeros on system rays only ρ) 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) − ih′ (φ)/ρ indicator f.