Perhaps this paper of Harris is what you are looking for: http://www.numdam.org/item?id=ASNSP_1981_4_8_1_35_0A bound on the geometric genus of projective varieties. It generalizes Castelnuovo's bound to smooth projective varieties of arbitrary dimension. In page 44, we find the following statement:
Theorem. Let $V$ be a non-degenerated smooth and irreducible variety of dimension $k$ and degree $d$ in $\mathbb P^n$. If we set $M=\left[ > \frac{d-1}{n-k}\right]$ then $$ > h^0(V,\Omega^k_V) \le \binom{M}{k+1} (n-k) + \binom{M}{k} (d-1-M(n-k)). $$ In particular, if $d \le k(n-k) + 1$ then $h^0(V, \Omega^k_V)=0.$