  We analyse two special cases of $\mu$-synthesis problems which can be reduced
to interpolation problems in the set of analytic functions from the disc into
the symmetrised bidisc and into the tetrablock. For these inhomogeneous domains
we study the structure of interconnections between the set of analytic
functions from the disc into the given domain, the matricial Schur class, the
Schur class of the bidisc, and the set of pairs of positive kernels on the
bidisc subject to a boundedness condition. We use the theories of Hilbert
function spaces and of reproducing kernels to establish these connections. We
give a solvability criterion for the interpolation problem that arises from the
$\mu$-synthesis problem related to the tetrablock.
