In this paper we consider the class ofsurface mappings consisting of those maps which have the least number offixed points possible among all maps in their homotopy class.When the surface has non-empty boundary, we show that for mappingsin this class the index of a fixed point is bounded above by $1$and below by $2\chi -1$.This generalizes a well known result for pseudo-Anosov homeomorphisms.A proof of a Jiang–Guo type inequality is also given.