A general vanishing theorem

Let E be a vector bundle and L be a line bundle over a smooth projective variety X. In this article, we give a condition for the vanishing of Dolbeault cohomology groups of the form $H^{p,q}(X,{\mathcal S}^{\alpha}E\otimes \wedge^{\beta} E\otimes L)$ when S ^α?+?β E???L is ample. This condition is shown to be invariant under the interchange of p and q. The optimality of this condition is discussed for some parameter values.