Consider a function $f(x,\lambda):\mathbb{R}^{2}_{+}\to\mathbb{R}_{+}$ that is uniformly continuous, smooth, lower bounded and convex. Let
$\qquad g(\lambda)=\inf_{x}\;f(x,\lambda)$
We know that $g(0)=0$, $g^{'}(\lambda)>0$, and $g(a\lambda)<ag(\lambda), \forall a>1$. Can we conclude that $g$ is also concave?
