$f$ convex $\Rightarrow$ $L(f)\geq f$

Assume that $L:C[a,b]\to C[a,b]$ is a sequence of linear operator acting on the set $C[a,b]$ of continuous functions over [a,b]. Besides, $L(1)=1$ and $L(x)=x$. I recently found that from this is trivial that $L(f)\geq f$ in $[a,b]$. But I cant see it. Any help is welcome. Thanks in advance.