  For a given domain $D$ in the extended complex plane $\bar{\mathbb C}$ with
an accessible boundary point $z_0 \in \partial D$ and for a subset $E \subset
{D},$ relatively closed w.r.t. $D,$ we define the relative capacity $\rc E$ as
a coefficient in the asymptotic expansion of the Ahlfors-Beurling conformal
invariant $r(D\setminus E,z)/r(D, z)$ when $z$ approaches the point $z_0.$ Here
$r(G,z)$ denotes the inner radius at $z$ of the connected component of the set
$G$ containing the point $z.$ The asymptotic behavior of this quotient is
established. Further, it is shown that in the case when the domain $D$ is the
upper half plane and $z_0=\infty$ the capacity $\rc E$ coincides with the
well-known half-plane capacity ${\hc} E.$ Some properties of the relative
capacity are proven, including the behavior of this capacity under various
forms of symmetrization and under some other geometric transformations. Some
applications to bounded holomorphic functions of the unit disk are given.
