  We give a simple construction of the log-convex minorant of a sequence
$\{M_\alpha\}_{\alpha\in\mathbb{N}_0^d}$ and consequently extend to the
$d$-dimensional case the well-known formula that relates a log-convex sequence
$\{M_p\}_{p\in\mathbb{N}_0}$ to its associated function $\omega_M$, that is
$M_p=\sup_{t>0}t^p\exp(-\omega_M(t))$. We show that in the more dimensional
anisotropic case the classical log-convex condition $M_\alpha^2\leq
M_{\alpha-e_j}M_{\alpha+e_j}$ is not sufficient: convexity as a function of
more variables is needed (not only coordinate-wise). We finally obtain some
applications to the inclusion of spaces of rapidly decreasing
ultradifferentiable functions in the matrix weighted setting.
