  Costa's entropy power inequality is an important generalization of Shannon's
entropy power inequality. Related with Costa's entropy power inequality and a
conjecture proposed by McKean in 1966, Cheng-Geng recently conjectured that
$D(m,n): (-1)^{m+1}(\partial^m/\partial^m t)H(X_t)\ge0$, where $X_t$ is the
$n$-dimensional random variable in Costa's entropy power inequality and
$H(X_t)$ the differential entropy of $X_t$. $D(1,n)$ and $D(2,n)$ were proved
by Costa as consequences of Costa's entropy power inequality. Cheng-Geng proved
$D(3,1)$ and $D(4,1)$. In this paper, we propose a systematical procedure to
prove $D(m,n)$ and Costa's entropy power inequality based on semidefinite
programming. Using software packages based on this procedure, we prove $D(3,n)$
for $n=2,3,4$ and give a new proof for Costa's entropy power inequality. We
also show that with the currently known constraints, $D(5,1)$ and $D(4,2)$
cannot be proved with the procedure.
