Upper bounds on the genus of hyperelliptic Albanese fibrations

Let $S$ be a minimal irregular surface of general type, whose Albanese map induces a hyperelliptic fibration $f:S\to B$ of genus $g$.We prove a quadratic upper bound on the genus $g$, i.e., $g\le h(\chi ({\mathcal{O}}_S))$, where $h$ is a quadratic function. We also construct examples showing that the quadratic upper bounds can not be improved to the linear ones. In the special case when $p_g(S)=q(S)=1$, we show that $g\le 14$.