The trivial lower bound for the size of a maximal partial ovoid of
$H\left(3,{q}^{2}\right)$ is
${q}^{2}+1$.
Ebert showed that this bound can be attained if and only if
$q$
is even. In the present paper it is shown that a maximal partial ovoid of
$H\left(3,{q}^{2}\right)$,
$q$ odd, has at
least
${q}^{2}+1+\frac{4}{9}q$ points
(previously, only
${q}^{2}+3$
was known). It is also shown that a maximal partial spread of
$H\left(3,{q}^{2}\right)$,
$q$ even, has
size
${q}^{2}+1$ or size
at least
${q}^{2}+1+\frac{4}{9}q$.
