Let \({\left\{{I}_{k}\right\}}_{k\in {\mathbb{Z}}}\) , Ik = [ak, bk], bk < ak+1, ak → −∞ (k → −∞), ak → +∞ (k → +∞) be a set of disjoint segments of the real axis \({\mathbb{R}}\) . Jk = [bk, ak+1], \(E=\bigcup_{k\in {\mathbb{Z}}}{J}_{k}\) . We assume that a0 = −1, b0 = 1, a1 = \({2}^{{n}_{0}}\stackrel{\scriptscriptstyle\mathrm{def}}{=}C\) , b−1 = \({-2}^{{n}_{0}}\) , |Ik| = 2−mα, α > 0 in case Ik ⊂ [2m, 2m+1] or Ik ⊂ [−2m+1,−2m], m ≥ n0. We assume further that there exist k and l such that ak = 2n and bl = −2n, for any n ≥ n0. The B. Ya. Levin function fE,σ(z), σ > 0, is defined to be a function satisfying the following conditions:
1) fE,σ(z) is subharmonic on the complex plane \({\mathbb{C}}\) and harmonic on \({\mathbb{C}}\backslash E\) ;
2) fE,σ(z) = 0, x ∈ E; fE,σ(z) > 0, z ∈ \({\mathbb{C}}\backslash E\) ;
3) \(\underset{z\to \infty }{\overline{\mathrm{lim}}}\frac{{f }_{E,\sigma }\left(z\right)}{\left|z\right|}=\sigma \) , \({f}_{E,\sigma }\left(\overline{z }\right)\) = \({f}_{E,\sigma }\left(z\right)\) ;
4) if g is subharmonic on \({\mathbb{C}}\) , g(x) ≤ 0, x ∈ E, and \(\underset{z\to \infty }{\overline{\mathrm{lim}} }\frac{g\left(z\right)}{\left|z\right|}\le \sigma \) , then \(\begin{array}{cc}g\left(z\right)\le {f}_{E,\sigma }\left(z\right),& z\in {\mathbb{C}}\end{array}.\)
The B. Ya. Levin function fE,σ(z) exists if C1|Il| ≥ |Jk| ≥ C|Il|, Jk, Il ⊂ [2n, 2n+1] or Jk, Il ⊂ [−2n+1,−2n], n ≥ n0. We prove that if C ≥ c0(α), then \(\underset{x\in {I}_{k}}{\mathrm{max}}{f}_{E,\sigma }\left(x\right)\le 6\sigma \left|{I}_{k}\right|\) and describe the behavior of fE,1(z) in a neighborhood of Jk, k ∈ \({\mathbb{Z}}\) .
