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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
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