  Let $T_{\epsilon}$, $0 \le \epsilon \le 1/2$, be the noise operator acting on
functions on the boolean cube $\{0,1\}^n$. Let $f$ be a distribution on
$\{0,1\}^n$ and let $q > 1$. We prove tight Mrs. Gerber-type results for the
second Renyi entropy of $T_{\epsilon} f$ which take into account the value of
the $q^{th}$ Renyi entropy of $f$. For a general function $f$ on $\{0,1\}^n$ we
prove tight hypercontractive inequalities for the $\ell_2$ norm of
$T_{\epsilon} f$ which take into account the ratio between $\ell_q$ and
$\ell_1$ norms of $f$.
