  We investigate the relation between the topological entropy of pseudo-Anosov
maps on surfaces with punctures and the rank of the first homology of their
mapping tori. On the surface $S$ of genus $g$ with $n$ punctures, we show that
the entropy of a pseudo-Anosov map is bounded from above by
$\dfrac{(k+1)\log(k+3)}{|\chi(S)|}$ up to a constant multiple when the rank of
the first homology of the mapping torus is $k+1$ and $k, g, n$ satisfy a
certain assumption. This is a partial generalization of precedent works of Tsai
and Agol-Leininger-Margalit.
