Let \(A\) be the class of plane curves \(z = z(t)\) without self-intersections such that \(z(0)= 0\), \(\lim_{t \to \infty} z(t) = \infty\), for any \(T \in (0, \infty)\) the curve \(z = z(t)\), \(t \in [0,T]\) be a union of a finite number of line segments. Let \({\mathcal L}_1\), \({\mathcal L}_2 \in A\) and let \(D\) be the open set such that \(\partial D = {\mathcal L}_1 \cup {\mathcal L}_2\). For holomorphic functions \(f\) in \(D\) the value \(M(r,D,f)\) is defined as \[ M(r,D,f) = \sup \biggl\{ \bigl |f (re^{i \theta}) \bigr |: re^{i \theta} \in D \biggr\}. \] The authors prove: Theorem. Let \({\mathcal L}_1\), \({\mathcal L}_2 \in A\), \(f\) be holomorphic in \(D\) and continuous up to the boundary. Let \(\eta \in (0, \pi)\) and the angular measure of the intersection \(D \cap \{z : |z |= r\}\) does not exceed \(2 \eta\). Let \(a(z)\) and \(b(z)\) be distinct entire functions of order strictly less then \(1/(2 + 2 \eta/ \pi)\). Let \(\lim_{{z \to \infty \atop z \in {\mathcal L}_1}} (f(z) - a(z)) = 0\), \(\lim_{{z \to \infty \atop z \in {\mathcal L}_2}} (f(z) - b(z)) = 0\). Then \(\varliminf_{z \to \infty} {\log M (r,D,f) \over r^{\pi/2 \eta}} > 0\). Earlier (1995) such type theorem with \({\mathcal L}_1\) and \({\mathcal L}_2\) being the rays was proved by Dudley Ward and Fenton.