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