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Plane indicator diagram of an entire function of an integer order \(\rho >1\). (English. Russian original) Zbl 0627.30022

Sib. Math. J. 28, No. 1-2, 263-277 (1987); translation from Sib. Mat. Zh. 28, No. 2(162), 107-123 (1987).
The well-known theorem of G. Pólya states that given an entire function \(f(z)=\sum_{k\geq 0}c_ kz^ k\) of order \(\rho =1\) and of normal type there exists the smallest convex compact K, outside of which the Borel transform \[ f(z)=\sum_{k\geq 0}c_ kk!z^{-k-1} \] is analytic. The support function \(k(\theta)\) of K is equal to \(h(-\theta)\) where \(h(\theta)\) is the indicator of f: \[ h(\theta)=\lim_{r\to \infty}r^{-1} \ell n| f(re^{i\theta})|. \] V. Bernstein in 1936 constructed a conjugate diagram K on a many-sheeted surface for an entire function f of order \(\rho >1\) and of normal type. The author proposes a plane analogue of a conjugate diagram of an entire function of order \(\rho >0\) and of normal type.
Reviewer: V.A.Tkachenko

MSC:

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