Let $\lambda \in ]0,1[$ and $f, g, h$ be functions from $\mathbb{R}^{2}$ to $\mathbb{R}_{+}$ that are continuous with bounded support and such that
$$\forall X \in \mathbb{R}^{2}, \forall Y \in \mathbb{R}^{2}, \quad h(\lambda X + (1-\lambda) Y) \geq f(X)^{\lambda} g(Y)^{1-\lambda}$$
Show that
$$\iint_{\mathbb{R}^{2}} h(x,y)\,dx\,dy \geq \left(\iint_{\mathbb{R}^{2}} f(x,y)\,dx\,dy\right)^{\lambda} \left(\iint_{\mathbb{R}^{2}} g(x,y)\,dx\,dy\right)^{1-\lambda}.$$