  Ishii and Sands showed the monotonicity of the Lozi family $\mathcal L_{a,b}$
in a $\mathcal C^1$ neighborhood of a-axis in the $a$-$b$ parameter space. We
show the monotonicity of the entropy in the vertical direction around $a=2$ and
in some other directions for $1<a\leq2$. Also we give some rigorous and
numerical results for the parameters at which the Lozi family has zero entropy.
