5

Prove that if $f,g:[0,1]\rightarrow[0,1]$ - continuous functions and f is strictly increasing then $$\int\limits_0^1f(g(x))dx\leq\int\limits_0^1f(x)dx+\int\limits_0^1g(x)dx.$$

I tried to prove that $f(g(x))\leq f(x)+g(x)$ for all $x\in[0,1]$ but realized that it is wrong. For example, $f(x)=\begin{cases}0,0\leq x< \frac{1}{4}\\2x-\frac{1}{2},\frac{1}{4}\leq x < \frac{3}{4} \\1,\frac{3}{4}\leq x\leq 1\end{cases}$, $g(x)=\frac{3}{4}$ but $f(g(0))=1>0+\frac{3}{4}=f(0)+g(0)$.

Martin Sleziak
  • 56,060
user122380
  • 177

1 Answers1

9

Let $x_0$ the point of $[0,1]$ where $h(x)=f(x)-x$ reach the maximum value.

We have $$ \int\limits_0^1f(x)dx \geq \int\limits_{x_0}^{1} f(x)dx \geq (1-x_0)f(x_0) $$ $$ =h(x_0)+x_0(1-f(x_0))\geq h(x_0)=max_{[0,1]} h(x) $$ $$ \geq \int\limits_{0}^{1} h(g(x))dx=\int\limits_0^1f(g(x))dx -\int\limits_0^1 g(x)dx $$