  Let $S_n$ be a punctured Riemann spheres $\mathbf{S}^2\backslash \{x_1,...,
x_n\}$. In this paper, we investigate pseudo-Anosov maps on $S_n$ that are
isotopic to the identity on $S_n\cup \{x_n\}$ and have the smallest possible
dilatations. We show that those maps cannot be obtained from Thurston's
construction (that is the products of two Dehn twists). We also prove that
those pseudo-Anosov maps $f$ on $S_n$ with the minimum dilatations can never
define a trivial mapping class as any puncture $x_i$ of $S_n$ is filled in. The
main tool is to give both lower and upper bounds estimations for dilatations
$\lambda(f)$ of those pseudo-Anosov maps $f$ on $S_n$ isotopic to the identity
as a puncture $x_i$ of $S_n$ is filled in.
