  In this paper, we study the class of relatively $D$-stable matrices and
provide the conditions, sufficient for relative $D$-stability. We generalize
the well-known Hadamard inequality, to provide upper bounds for the
determinants of relatively $D$-stable and relatively additive $D$-stable
matrices. For some classes of $D$-stable matrices, we estimate the sector gap
between matrix spectra and the imaginary axis. We apply the developed technique
to obtain upper bounds for determinants of some classes of $D$-stable matrices,
e.g. diagonally stable, diagonally dominant and matrices with $Q^2$-scalings.
