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 Algebra i Analiz: Year: Volume: Issue: Page: Find

 Algebra i Analiz, 1998, Volume 10, Issue 3, Pages 31–44 (Mi aa995)

Research Papers

Lower bounds on the values of an entire function of exponential type at certain integers, in terms of a least superharmonic majorant

P. Koosisa, Henrik L. Pedersenb

a Mathematics Department, McGill University, Montreal, Québec, Canada
b Matematisk Afdeling, Københavns Universitet, København, Denmark

Abstract: In this,paper and the following one, it is shown that if $A<\pi$ and $\eta>0$ is sufficiently small (depending on $A$), the entire functions $f(z)$ of exponential type $\le A$ satisfying $\sum^{\infty}_{m=-\infty}(\log^+|f(n)|/(1+n^2))\le\eta$ form a normal family (in $\mathbb C$). General properties of least superharmonic majorants are used to obtain this result, and from it the multiplier theorem of Beurling and Malliavin is readily derived.

Keywords: Entire function of exponential type, least superharmonic majorant, logarithmic sum, BeurlingT-Malliavin multiplier theorem.

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English version:
St. Petersburg Mathematical Journal, 1999, 10:3, 429–439

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Citation: P. Koosis, Henrik L. Pedersen, “Lower bounds on the values of an entire function of exponential type at certain integers, in terms of a least superharmonic majorant”, Algebra i Analiz, 10:3 (1998), 31–44; St. Petersburg Math. J., 10:3 (1999), 429–439

Citation in format AMSBIB
\Bibitem{KooPed98} \by P.~Koosis, Henrik L.~Pedersen \paper Lower bounds on the values of an entire function of exponential type at certain integers, in terms of a~least superharmonic majorant \jour Algebra i Analiz \yr 1998 \vol 10 \issue 3 \pages 31--44 \mathnet{http://mi.mathnet.ru/aa995} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1628018} \zmath{https://zbmath.org/?q=an:0920.30021} \transl \jour St. Petersburg Math. J. \yr 1999 \vol 10 \issue 3 \pages 429--439 

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This publication is cited in the following articles:
1. Pedersen H.L., “Entire functions and logarithmic sums over nonsymmetric sets of the real line”, Annales Academiae Scientiarum Fennicae–Mathematica, 25:2 (2000), 351–388
2. Koosis P., “Use of logarithmic sums to estimate polynomials”, Annales Academiae Scientiarum Fennicae–Mathematica, 26:2 (2001), 409–446
3. Pedersen H.L., “Estimates of entire functions of exponential type less than $\pi$ in terms of logarithmic sums over real Duffin and Schaeffer sequences”, Potential Anal., 19:3 (2003), 251–260
4. J. Mashreghi, F. L. Nazarov, V. P. Havin, “Beurling–Malliavin multiplier theorem: the seventh proof”, St. Petersburg Math. J., 17:5 (2006), 699–744
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