  In this paper, we use the semi-group method and an adaptation of the
$L^2-$method of H\"ormander to establish some $\Phi-$entropy inequalities and
asymmetric covariance estimates for the strictly convex measures in $\mathbb
R^n$. These inequalities extends the ones for the strictly log-concave measures
to more general setting of convex measures. The $\Phi-$entropy inequalities are
turned out to be sharp in the special case of Cauchy measures. Finally, we show
that the similar inequalities for log-concave measures can be obtained from our
results in the limiting case.
