SCHWEITZER 2001 Problem 6

Let $I\subset \mathbb R$ be a non-empty open interval, $\varepsilon\geq 0$ and $f\colon I\rightarrow\mathbb R$ a function satisfying the $$f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)+\varepsilon t(1-t)|x-y|$$ inequality for all $x,y\in I$ and $t\in [0,1]$ . Prove that there exists a convex $g\colon I\rightarrow\mathbb R$ function, such that the function $l :=f-g$ has the $\varepsilon$ -Lipschitz property, that is $$|l(x)-l(y)|\leq \varepsilon|x-y|\text{ for all }x,y\in I$$