  We study the asymptotic behavior of the Ablowitz-Segur solutions for the
second Painlev\'e equation using the Riemann-Hilbert approach and methods based
on asymptotic expansions of classical special functions. Recent results show
that the matrix-valued function satisfying the associated Riemann-Hilbert
problem can be represented by means of a local parametrix around the origin,
whose existence can be proved by a vanishing lemma. The aim of this paper is to
construct the explicit form of this parametrix and apply it to obtain improved
asymptotic relations for the real and purely imaginary Ablowitz-Segur solutions
of the inhomogeneous Painlev\'e II equation.
