Let $\{Z(t)\}_{t\geq 0}$ be a strongly continuous positive semigroup on a Banach lattice V. (endowed with ordering $\leq$), s.t for the identity element $1\in V$, $Z(t)1=1$, for all $t\geq 0$. $\phi:V\rightarrow V$ is a convex operator. I want to prove that $$\phi(Z(t)f)\leq Z(t)(\phi f),\quad f\in V,t>0$$
can someone out there help me out?