Crowding for analytic functions with elongated range

Abstract

Letf be a function analytic in the unit diskD. If the rangef(D) off is contained in a rectangleR with sidesa andb withb≤a such thatf(D) touches both small sides ofR, then the supremum norm of the derivative satisfies ‖f′‖ ≥b·ψ(b/a). We derive tight bounds for the best possible function ψ in this estimate. In particular, we show that$\psi (\tau ) \sim (2\pi \sqrt e )^{ - 1} \cdot \exp (\pi /2\tau )$ for small τ.