Let \(I_k=[a_k,b_k],\ J_k=[b_k,a_{k+1}],\ b_k<a_{k+1},\ k\in \mathbb {Z}\) , be segments on the real axis tending to \(+\infty \) and to \(-\infty \) . Assume that these segments satisfy the conditions \(|I_k|=2^{-n\alpha }\) if \(I_k\subset [2^{n},2^{n+1}] \) or \(I_k\subset [-2^{n+1},-2^{n}] \) , \(\alpha >0\) being fixed, \(n\ge n_0\) . Assume also that there exists a constant \(c_1>0\) such that \( 2^{n_0}\cdot 2^{-n\alpha }\le |J_k|\le c_1\cdot 2^{n_0}\cdot 2^{-n\alpha }\) if \(J_k\subset [2^{n},2^{n+1}]\) or \(J_k\subset [-2^{n+1},-2^{n}],k\in \mathbb {Z}\) . Put \(E=\bigcup \limits _{k\in \mathbb {Z}}J_k\) . Denote by \(f_{E,1}(z)\) a function subharmonic on \(\mathbb {C}\) and satisfying the conditions \(f_{E,1}(x)=0,x\in E, f_{E,1}(z)\) is harmonic on \(\mathbb {C}\setminus E\) , \( \underset{z\rightarrow \infty }{\varlimsup }\ \dfrac{f_{E,1}(z)}{|z|}=1\) , and \(g(z)\le f_{E,1}(z),\ z\in \mathbb {C}\) , for every function g subharmonic on \(\mathbb {C}\) and such that \(g(x)\le 0\) , \(x\in E\) , and \( \underset{z\rightarrow \infty }{\varlimsup }\dfrac{g(z)}{|z|}\le 1.\) We define sets \(L_t(E)\) as follows: \(\begin{aligned} L_t=\{z\in \mathbb {C}: f_{E,1}(z)=t\} \end{aligned}\) and put \(\rho _t(x)= \textrm{dist}(x,L_t(E)),\ x\in E\) . Let \(T_{\sigma }\) be the set of entire functions of exponential type satisfying the condition \(|F_{\sigma }(z)|\le c_{F_{\sigma }}\exp (\sigma |\text {Im}z|),z\in \mathbb {C}, F_{\sigma }\in T_{\sigma }.\) Denote by \(\Lambda ^s(E)\) the s-Hölder class on \(E,\ 0<s<1,\) of functions bounded on the set E. We prove the following result. Theorem 1. Assume that for a function f defined on E there exist functions \(F_{\sigma }\in T_{\sigma }\) such that \(\begin{aligned} |f(x)-F_{\sigma }(x)|\le c_f\rho ^s_{\frac{1}{\sigma }}(x),\quad x\in E,\quad \sigma \ge 1. \end{aligned}\) Then \(f \in \Lambda ^s(E)\) . Bibliography: 7 titles.
