Direct images of bundles under Frobenius morphism

Let X be a smooth projective variety of dimension n over an algebraically closed field k with char(k)=p>0 and F:X→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⊗T^ℓ(Ω¹ _X ) (0≤ℓ≤n(p-1)) (Corollary 4.9). When X is a smooth projective curve of genus g≥2, it implies F _* W being stable whenever W is stable. Dedicated to Professor Zhexian Wan on the occasion of his 80th birthday.