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On sets of regular increase of entire functions. I. (Russian) Zbl 0601.30036
Let $$\rho$$ (r) ($$\rho$$ (r)$$\to \rho$$ as $$r\to \infty)$$ be a proximate order, H($$\theta)$$ a $$2\pi$$-periodic $$\rho$$-trigonometrically convex function, [$$\rho$$ (r),H($$\theta)$$] the class of entire functions with $\limsup_{r\to \infty}r^{-\rho (r)} \ln | f(re^{i\theta})| \leq H(\theta)$ for all $$\theta$$, $$\Lambda:=\{\lambda_ n:$$ $$...\leq | \lambda_ n| \leq | \lambda_{n+1}| \leq...\to \infty$$, $$\lambda_ i\neq \lambda_ j$$ for $$i\neq j\}$$ and $A:=\{\{a_ n\}: \limsup_{n\to \infty}| \lambda_ n|^{-\rho (| \lambda_ n|)} \ln | a_ n| -H(\arg \lambda_ n)\leq 0\}.$ $$\Lambda$$ is called an interpolation set in [$$\rho$$ (r),H($$\theta)$$] with respect to A if for each sequence $$\{a_ n\}\in A$$ there exists f(x)$$\in [\rho (r),H(\theta)]$$ such that $$f(\lambda_ n)=a_ n$$ for all $$n\geq 1.$$
The paper is divided into three parts and establishes criteria for the interpolation property of the set $$\Lambda$$. In this part, the author obtains auxiliary results. Earlier, the reviewer had established this result in [$$\rho$$ (r),H($$\theta)$$], under a certain condition [see: Mechanics of a continuous medium (Russian), 49-54, Rostov, Gos. Univ., Rostov-on-Don, 1981; RZH-Mat 1981:8B136].

##### MSC:
 30D15 Special classes of entire functions of one complex variable and growth estimates