  Let $f: S\longrightarrow B$ be a non-trivial fibration from a complex
projective smooth surface $S$ to a smooth curve $B$ of genus $b$. Let $c_f$ the
Clifford index of the generic fibre $F$ of $f$. In [arXiv:1401.7502v4] it is
proved that the relative irregularity of $f$, $q_f=h^{1,0}(S)-b$ is less than
or equal to $g(F)-c_f$. In particular this proves the (modified) Xiao's
conjecture: $q_f\le 1+g(F)/2$ for fibrations of general Clifford index. In this
short note we assume that the generic fiber of $f$ is a plane curve of degree
$d\ge 5$ and we prove that $q_f\le g(F)-c_f-1$. In particular we obtain the
conjecture for families of quintic plane curves. This theorem is implied for
the following result on infinitesimal deformations: let $F$ a smooth plane
curve of degree $d\ge 5$ and let $\xi$ be an infinitesimal deformation of $F$
preserving the planarity of the curve. Then the rank of the cup-product map
$\cdot \xi: H^0(F,\omega_F) \rightarrow H^1(F,O_F)$ is at least $d-3$. We also
show that this bound is sharp.
