  Suppose $\pi : X \to Y$ is a smooth blow-up along a submanifold $Z$ of $Y$
between complex Fano manifolds $X$ and $Y$ of pseudo-indices $i_X$ and $i_Y$
respectively (recall that $i_X$ is defined by $i_X := \min \{-K_X \cdot C | C
{is a rational curve of} X\}$). We prove that $i_X \leq i_Y$ if $2\dim (Z)<
\dim(Y) + i_Y -1$ and show that this result is optimal by classifying the
``boundary'' cases. As expected, these results are obtained by studying
rational curves on $X$ and $Y$.
