  This paper concerns the evolution of a closed convex hypersurface in
${\mathbb{R}}^{n+1}$, in direction of its inner unit normal vector, where the
speed is given by a smooth function depending only on the mean curvature, and
satisfies some further restrictions, without requiring homogeneity. It is shown
that the flow exists on a finite maximal interval, convexity is preserved and
the hypersurfaces shrink down to a single point as the final time is
approached. This result covers and generalises the corresponding result of
Schulze \cite{Sch05} for the positive power mean curvature flow to a much
larger possible class of flows by the functions depending only on the mean
curvature.
