Let $f: \mathbb R_{+} \to \mathbb R_{+}$ be an $s$-function in the second sense, i.e., $$ f(\lambda x +(1-\lambda)y) \leq \lambda^s f(x) +(1-\lambda)^s f(y)$$ for every $\lambda \in (0,1)$. Assume further that, there exists a convex function $g: \mathbb R_{+} \to \mathbb R_{+}$ and $a< b \in \mathbb R_{+}$ such that $$ a g(x) \leq f(x) \leq b g(x).$$ Is it true that $f$ is also convex? Or any criterion so that $f$ is convex? One trivial example is $f$ should have the form $f= c g$ for $a\leq c \leq b$, but in general I cannot figure out any other condition for convexity.
