On an estimate for a Dirichlet polynomial and some of its applications

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.
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