  We obtain an improvement of the Beckner's inequality $\| f\|^{2}_{2}
-\|f\|^{2}_{p} \leq (2-p) \| \nabla f\|_{2}^{2}$ valid for $p \in [1,2]$ and
the Gaussian measure. Our improvement is essential for the intermediate case $p
\in (1,2)$, and moreover, we find the natural extension of the inequality for
any real $p$.
