  Let $f:X\rightarrow Z$ be a separable fibration of relative dimension 1
between smooth projective varieties over an algebraically closed field $k$ of
positive characteristic. We prove the subadditivity of Kodaira dimension
$\kappa(X)\geq\kappa(Z)+\kappa(F)$, where $F$ is the generic geometric fiber of
$f$, and $\kappa(F)$ is the Kodaira dimension of the normalization of $F$.
Moreover, if $\dim X=2$ and $\dim Z=1$, we have a stronger inequality
$\kappa(X)\geq \kappa(Z)+\kappa_1(F)$ where $\kappa_1(F)=\kappa(F,\omega^o_F)$
is the Kodaira dimension of the dualizing sheaf $\omega_F^o$.
