Abstract:
The idea to write this note arose during the discussion that followed the report of the first author at the International Scientific Conference “Ufa Autumn Mathematical School-2022”. We propose three general methods for constructing uniqueness sets in classes of entire functions with growth restrictions. In all three cases, the sequence of zeros of an entire function with special properties is chosen as such set. The first method is related to Sylvester famous problem on the smallest circle containing a given set of points on a plane, and theorems of convex geometry. The second method initially relies on Helly theorem on the intersection of convex sets and its application to the possibility of covering one set by shifting another. The third method is based on the classical Jensen formula, which allows one to estimate the type of an entire function via the averaged upper density of the sequence of its zeros. We present only basic results now. The development of our approaches is expected to be presented in subsequent works.
Keywords:
Sylvester problem, Jung theorem, Helly theorem, uniqueness set, type of entire function, sequence of zeros, indicator of an entire function, averaged upper density, Jensen formula, indicator diagram, smallest circle.
The work by the second author is made in the framework of the state task of the Ministry of Science and
Higher Education of Russian Federation (code of scientific theme FMRS-2022-0124).
Citation:
G. G. Braichev, B. N. Khabibullin, V. B. Sherstyukov, “Sylvester problem, coverings by shifts, and uniqueness theorems for entire functions”, Ufa Math. J., 15:4 (2023), 31–41
\Bibitem{BraKhaShe23}
\by G.~G.~Braichev, B.~N.~Khabibullin, V.~B.~Sherstyukov
\paper Sylvester problem, coverings by shifts, and uniqueness theorems for entire functions
\jour Ufa Math. J.
\yr 2023
\vol 15
\issue 4
\pages 31--41
\mathnet{http://mi.mathnet.ru/eng/ufa674}
\crossref{https://doi.org/10.13108/2023-15-4-31}
Linking options:
https://www.mathnet.ru/eng/ufa674
https://doi.org/10.13108/2023-15-4-31
https://www.mathnet.ru/eng/ufa/v15/i4/p30
This publication is cited in the following 2 articles:
G. G. Braichev, “Zadacha Silvestra i mnozhestva edinstvennosti v klassakh tselykh funktsii”, Funktsionalnye prostranstva. Differentsialnye operatory. Problemy
matematicheskogo obrazovaniya, SMFN, 70, no. 1, Rossiiskii universitet druzhby narodov, M., 2024, 25–37
G. G. Braichev, V. B. Sherstyukov, “Uniqueness Theorem for Entire Functions of Exponential Type”, Lobachevskii J Math, 45:6 (2024), 2672
Citation in format AMSBIB
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