On the depth of the tangent cone and the growth of the Hilbert function

For a $d-$dimensional Cohen-Macaulay local ring $(R, \mathbf{m})$ we study the depth of the associated graded ring of $R$ with respect to an $\textbf{ m}$-primary ideal $I$ in terms of the Vallabrega-Valla conditions and the length of $I^{t+1}/JI^{t}$, where $J$ is a $J$ minimal reduction of $I$ and $t\ge 1$. As a corollary we generalize Sally's conjecture on the depth of the associated graded ring with respect to a maximal ideal to $\mathbf{m}$-primary ideals. We also study the growth of the Hilbert function.