  We consider a univalent analytic function $f$ on the half-plane satisfying
the condition that the supremum norm of its (pre-)Schwarzian derivative
vanishes on the boundary. Under certain extra assumptions on $f$, we show that
there exists a chordal Loewner chain initiated from $f$ until some finite time,
and this Loewner chain defines a quasiconformal extension of $f$ over the
boundary such that its complex dilatation is given explicitly in terms of the
(pre-)Schwarzian derivative in some neighborhood of the boundary. This can be
regarded as the half-plane version of the corresponding result developed on the
disk by Becker and also the generalization of the Ahlfors-Weill formula. As an
application of this quasiconformal extension, we complete the characterization
of an element of the VMO-Teichm\"uller space on the half-plane using the
vanishing Carleson measure condition induced by the (pre-)Schwarzian
derivative.
