  Let $W$ be a manifold with boundary $M$ given together with a conformal class
$\bar C$ which restricts to a conformal class $C$ on $M$. Then the relative
Yamabe constant $Y_{\bar C}(W,M;C)$ is well-defined. We study the short-time
behavior of the relative Yamabe constant $Y_{[\bar g_t]}(W,M;C)$ under the
Ricci flow $\bar g_t$ on $W$ with boundary conditions that mean curvature
$H_{\bar g_t}\equiv 0$ and $\bar{g}_t|_M\in C = [\bar{g}_0]$. In particular, we
show that if the initial metric $\bar{g}_0$ is a Yamabe metric, then, under
some natural assumptions, $\left.\frac{d}{dt}\right|_{t=0}Y_{[\bar
g_t]}(W,M;C)\geq 0$ and is equal to zero if and only the metric $\bar{g}_0$ is
Einstein.
