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Decomposition of entire functions of finite order into equivalent factors. (English. Russian original) Zbl 0671.30026

Transl., Ser. 2, Am. Math. Soc. 142, 61-72 (1989); translation from Problems in the approximation of functions of real and complex variables, Ufa, 161-181 (1983).
We describe a typical result of this paper. Let \(\Lambda =\{\lambda_ i\}\) and \(\Gamma =\{\gamma_ i\}\) be two given sequences of complex numbers tending to infinity \((\lambda_ i,\gamma_ i\neq 0)\). Put \[ G(z,p;\Lambda)=\prod_{| \lambda_ i| >0}G(z/\lambda_ i,p),\quad b_ k=\sum^{\infty}_{i=1}(1/\lambda^ k_ i-1/\gamma^ k_ i), \] and denote by n(t,\(\Lambda)\) the number of points from \(\Lambda\) lying in the closed disk of radius t. Now assume that \[ \sup_{t>0}n(t,\Lambda \cup \Gamma)/t^{\rho}<+\infty, \] for some \(\rho >0\) and \(| \lambda_ i-\gamma_ i| =O(| \lambda_ i| /| \lambda_ i|^{\rho \gamma})\) for some \(\gamma\in (0,1)\), \(i\to \infty\). Then to any number \(\alpha\in (0,1)\) there corresponds a set of circles \(E_{\alpha}\) (with certain linear density) such that \[ | \ell n| G(z,p;\Lambda)| -\ell n| G(z,p;\Gamma)| -\sum_{\rho (1-\gamma)\leq k\leq p}Re(b_ kz^ k\quad /k)| \leq \frac{const}{\alpha^ 2 \sin \pi \alpha}| z|^{(1-\gamma +\alpha)} \] holds for \(z\not\in E_{\alpha}\) and \(p=[\rho]\), \(| z| \to \infty.\)
{Reviewer’s remarks: See a book by B. Levin [Distribution of zeros of entire functions, Chapter II (1964; Zbl 0152.067)]. The printing of the paper under review is not neat and it makes reading difficult.}

MSC:

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

Citations:

Zbl 0152.067
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