Speaker
Description
Let $\{I_k\}_{k\in \mathbb{Z}}$ be a set of disjoint of segments of real axis, $E=\bigcup_{k\in \mathbb{Z}}{I_k}$. We call the B.Ya.Levin function $f_{E,\sigma}(z)$, $\sigma >0$, a function satisfying the following conditions:
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$f_{E,\sigma}(z)$ is subharmonic on the complex plane $\mathbb{C}$ and harmonic on $\mathbb{C}\setminus E$;
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$f_{E,\sigma}(z)=0$, $x\in E$; $f_{E,\sigma}(z)>0$, $z\in\mathbb{C} \setminus E$;
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$\limsup_{z\rightarrow \infty}\frac{f_{E,\sigma}(z)}{|z|}=\sigma$, $f_{E,\sigma}(\bar{z})=f_{E,\sigma}(z)$;
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If $g$ is subharmonic on $\mathbb{C}$, $g(x)\leq 0$, $x\in E$, $\limsup_{z\rightarrow \infty}\frac{g(z)}{|z|}\leq\sigma$, then $g(z)\leq f_{E,\sigma}(z)$.
We construct the B.Ya.Levin function for some sets of segments such that $|I_k|_{|k|\rightarrow \infty}\rightarrow 0$.
