On the existence of strips inside domains convex in one direction

A plane domain Ω is convex in the positive direction if for every ω ∈ Ω, the entire half-line {ω + t: t ≥ 0} is contained in Ω. Suppose that h maps the unit disk onto such a domain Ω with the normalization h(0) = 0 and lim_t→∞h^?1(h(z) + t) = 1. We show that if ∠lim_z→?1 Re h(z) = ?∞ and ∠lim_z→?1(1 + z)h′(z) = ν ∈ (0, +∞), then Ω contains a maximal horizontal strip of width πν. We also prove a converse statement. These results provide a solution to a problem posed by Elin and Shoikhet in connection with semigroups of holomorphic functions.