  Let $X$ be a smooth projective variety of dimension $n$ over an algebraically
closed field $k$ with ${\rm char}(k)=p>0$ and $F:X\to X_1$ be the relative
Frobenius morphism. For any vector bundle $W$ on $X$, we prove that instability
of $F_*W$ is bounded by instability of $W\otimes{\rm T}^{\ell}(\Omega^1_X)$
($0\le \ell\le n(p-1)$)(Corollary \ref{cor3.8}). When $X$ is a smooth
projective curve of genus $g\ge 2$, it implies $F_*W$ being stable whenever $W$
is stable.
