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 Izv. Akad. Nauk SSSR Ser. Mat., 1978, Volume 42, Issue 5, Pages 965–971 (Mi izv1886)

On the complete regularity of growth of the Borel transform of the analytic continuation of the associated function, which has a finite number of singular points

N. V. Govorov, N. M. Chernykh

Abstract: The following theorem is proved. Let $A(z)$ be an entire function of exponential type, and let its Borel transform $a(z)$ satisfy the following conditions: 1) $a(z)$ can be analytically continued to a certain Riemann surface $R$ with finite number of branch points, and it has only finitely many singularities $z_k$ on $R$; 2) in any plane with cuts by parallel rays issuing from the $z_k$, a branch of $z_k$ satisfies
$$\varlimsup_{z\to\infty}\frac{\ln|a(z)|}{|z|}\leq0.$$

Then $A(z)$ has completely regular growth. From this theorem it follows, in particular, that if $a(z)$ is an algebraic function or a single-valued function with a finite number of singularities, then $A(z)$ has completely regular growth.
Bibliography: 6 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1979, 13:2, 253–259

Bibliographic databases:

UDC: 517.53
MSC: 30D15

Citation: N. V. Govorov, N. M. Chernykh, “On the complete regularity of growth of the Borel transform of the analytic continuation of the associated function, which has a finite number of singular points”, Izv. Akad. Nauk SSSR Ser. Mat., 42:5 (1978), 965–971; Math. USSR-Izv., 13:2 (1979), 253–259

Citation in format AMSBIB
\Bibitem{GovChe78} \by N.~V.~Govorov, N.~M.~Chernykh \paper On the complete regularity of growth of the Borel transform of the analytic continuation of the associated function, which has a~finite number of singular points \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1978 \vol 42 \issue 5 \pages 965--971 \mathnet{http://mi.mathnet.ru/izv1886} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=513909} \zmath{https://zbmath.org/?q=an:0426.30022|0395.30019} \transl \jour Math. USSR-Izv. \yr 1979 \vol 13 \issue 2 \pages 253--259 \crossref{https://doi.org/10.1070/IM1979v013n02ABEH002042} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1979JD23800003} 

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This publication is cited in the following articles:
1. N. V. Govorov, N. M. Chernykh, “Complete regularity of growth for some classes of entire functions of exponential type represented by Âorel integrals and power series”, Math. USSR-Izv., 27:3 (1986), 431–450
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