Abstract:
It is proved that for any entire function f of finite nonzero order there is a set S in the plane with density zero and such that for any a∈C almost all the roots of the equation f(z)=a belong to S. This assertion was deduced by Littlewood from an unproved conjecture about an estimate of the spherical derivative of a polynomial. This conjecture is proved here in a weakened form.
Bibliography: 11 titles.
Citation:
A. È. Eremenko, M. L. Sodin, “Proof of a conditional theorem of Littlewood on the distribution of values of entire functions”, Math. USSR-Izv., 30:2 (1988), 395–402
\Bibitem{EreSod87}
\by A.~\`E.~Eremenko, M.~L.~Sodin
\paper Proof of a~conditional theorem of Littlewood on the distribution of values of entire functions
\jour Math. USSR-Izv.
\yr 1988
\vol 30
\issue 2
\pages 395--402
\mathnet{http://mi.mathnet.ru/eng/im1302}
\crossref{https://doi.org/10.1070/IM1988v030n02ABEH001020}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=897006}
\zmath{https://zbmath.org/?q=an:0638.30029|0627.30025}
Linking options:
https://www.mathnet.ru/eng/im1302
https://doi.org/10.1070/IM1988v030n02ABEH001020
https://www.mathnet.ru/eng/im/v51/i2/p421
This publication is cited in the following 2 articles:
John L. Lewis, “On a conditional theorem of Littlewood for quasiregular entire functions”, J Anal Math, 62:1 (1994), 169
A. È. Erëmenko, “Lower estimate in Littlewood's conjecture on the mean spherical derivative of a polynomial and iteration theory”, Proc. Amer. Math. Soc., 112:3 (1991), 713
Citation in format AMSBIB
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