Prove the following integral inequality: $\int_{0}^{1}f(g(x))dx\le\int_{0}^{1}f(x)dx+\int_{0}^{1}g(x)dx$

Question

Suppose $f(x)$ and $g(x)$ are continuous function from $[0,1]\rightarrow [0,1]$, and $f$ is monotone increasing, then how to prove the following inequality: $$\int_{0}^{1}f(g(x))dx\le\int_{0}^{1}f(x)dx+\int_{0}^{1}g(x)dx$$

Answer 1

By the Mean Value Theorem for Integrals, there is a point $\xi \in [0,1]$ such that $$ \int_{0}^{1}[f(g(x))-g(x)]dx=f(g(\xi))-g(\xi).\tag{1} $$ Let $u = g(\xi)$. Then $$ f(g(\xi))-g(\xi) = f(u)-u \leq f(u) - uf(u) = (1-u)f(u),\tag{2} $$ (the inequality is due to the fact that $0\leq f(u)\leq 1$). Since $f$ is monotone increasing, $f(x)\geq f(u)$ for all $x$ in $[u,1]$. So $$ (1-u)f(u) = \int_u^1 f(u)\,dx \leq \int_u^1 f(x)\,dx \leq \int_0^1 f(x)\,dx.\tag{3} $$ Hence, by $(1),(2),(3)$, we have
$$\int_{0}^{1}[f(g(x))-g(x)]dx\leq\int_0^1 f(x)\,dx.$$ Rearranging, $$ \int_0^1 f(g(x)) \,dx \leq \int_0^1 f(x)\,dx + \int_0^1 g(x)\,dx. $$ Over!