  This is a commentary on Teichm{\"u}ller's paper Ein Verschiebungssatz der
quasikonformen Abbildung (A displacement theorem of quasiconformal mapping),
published in 1944. We explain in detail how Teichm{\"u}ller solves the problem
of finding the quasiconformal mapping from the unit disc to itself, sending 0
to a strictly negative point on the real line, holding the boundary of the disc
pointwise fixed and with the smallest quasiconformal dilatation. We mention
also some consequences of this extremal problem and we ask a question.
