Albanese fibrations of surfaces with low slope

Let $S$ be a minimal irregular surface of general type, whose Albanese map induces a fibration $f:\,S \to C$ of genus $g$.We prove a linear upper bound on the genus $g$ if $K_S^2\leq 4\chi(\mathcal{O}_S)$. Examples are constructed showing that the above linear upper bound is sharp. We also give a characterization of the Albanese fibrations reaching the above upper bound when $\chi(\mathcal{O}_S)\geq 5$.On the other hand, we will construct a sequence of surfaces $S_n$ of general type with $K_{S_n}^2/\chi(\mathcal{O}_{S_n})>4$ and with an Albanese fibration $f_n$, such that the genus $g_n$ of a general fiber of $f_n$ increases quadratically with $\chi(\mathcal{O}_{S_n})$,and that $K_{S_n}^2/\chi(\mathcal{O}_{S_n})$ can be arbitrarily close to $4$.