An estimate for crowding in conformal mapping to elongated regions

There exists a function ψ(r) which is ∼(r/2π) exp (π/2r) for 0<r1 with the following property: If f is a function analytic in the unit dise such that its range w(f) is contained in a rectangle R with sides a and b b≤a, and W(f) touches both small sides of R, then supremum norn of the derivative satisfies . Examples show that this estimate is best possible up to a factor of about 2.