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On an estimate for a Dirichlet polynomial and some of its applications
A. F. Leont'ev
Abstract:
Let $L(\mu)$ be an entire function of exponential type and of completely regular growth, $\overline D$ its conjugate diagram, and $\overline D(\alpha)$ the displacement of $\overline D$ by the vector $\alpha$. Next let $\alpha_1$ and $\alpha_2$ be arbitrary fixed points, and $D_1$ and $D_2$ be regions such that $D_1\supset\overline D(\alpha_1)$ and $D_2\supset\overline D(\alpha_2)$. The estimate
$$
|P(z)|\leqslant N\max(M_1,M_2),\qquad M_j=\max_{t\in\overline D_j}|P(t)|\quad(j=1,2),
$$
where $N$ does not depend on $P(z)$, is established for a Dirichlet polynomial $P(z)$, whose exponents are the zeros of $L(\mu)$, in some region $G$ containing the set $\overline D(\alpha)$, $\alpha\in[\alpha_1,\alpha_2]$. A number of corollaries follow from the estimate.
Bibliography: 7 titles.
Received: 15.01.1973
Citation:
A. F. Leont'ev, “On an estimate for a Dirichlet polynomial and some of its applications”, Mat. Sb. (N.S.), 91(133):4(8) (1973), 554–564; Math. USSR-Sb., 20:4 (1973), 575–586
Linking options:
https://www.mathnet.ru/eng/sm3326https://doi.org/10.1070/SM1973v020n04ABEH001997 https://www.mathnet.ru/eng/sm/v133/i4/p554
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Abstract page: | 224 | Russian version PDF: | 100 | English version PDF: | 14 | References: | 46 |
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