A Berezin-Li-Yau type inequality for $(-\Delta)^{\alpha/2}|_{\Omega},$ the
fractional Laplacian operators restriced to a bounded domain $\Omega\subset
\mathbb{R}^d$ for $\alpha\in(0,2],$ $d\ge 2,$ has not been known so far. First
we positively answer this question. Second, we provide an improvement to this
inequality consistent with the work in \cite{Melas,Selma} by using a pure
analytical approach.