Sufficient condition for convexity involving an average of slopes

Question

Suppose $h(x)\ge0$ is increasing and concave for all $x\ge0$. For $\Delta>0$, let $$ f(x)=\frac{h(x+\Delta)-h(x)}{\Delta}. $$ I feel that $f$ is a convex function, i.e. $$ f(tx+(1-t)y)\le t f(x)+(1-t)f(y), 0<t<1. $$ But, I don't have any good progress on proving it. Is it obvious true? Or, is there any counterexample? Any reference, help or adivce would be of great help. Thanks.