  An information theoretic approach to bounds in superconformal field theories
is proposed. It is proved that the supersymmetric R\'enyi entropy $\bar
S_\alpha$ is a monotonically decreasing function of $\alpha$ and
$(\alpha-1)\bar S_\alpha$ is a concave function of $\alpha$. Under the
assumption that the thermal entropy associated with the "replica trick" time
circle is bounded from below by the charge at $\alpha\to\infty$, it is further
proved that both ${\alpha-1\over \alpha}\bar S_\alpha$ and $(\alpha-1)\bar
S_\alpha$ monotonically increase as functions of $\alpha$. Because $\bar
S_\alpha$ enjoys universal relations with the Weyl anomaly coefficients in
even-dimensional superconformal field theories, one therefore obtains a set of
bounds on these coefficients by imposing the inequalities of $\bar S_\alpha$.
Some of the bounds coincide with Hofman-Maldacena bounds and the others are
new. We also check the inequalities for examples in odd-dimensions.
