Let  be a punctured Riemann sphere  In this paper, we investigate pseudo-Anosov maps on  that are isotopic to the identity on  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  with the minimum dilatations can never define a trivial mapping class as any puncture  of  is filled in. The maintool is to give both lower and upper bounds estimations for dilatations  of those pseudo-Anosov maps f on  isotopic to the identity as a puncture  of  is filled in.