An Integral Inequality Involving Composite Functions
Inviato: 29/05/2019, 01:51
Problem:
Let $f,g:[0,1] -> [0,1]$ be continuous functions and $f$ be increasing.
Prove that:
\[
\int_0^1 f(g(x))\ \text{d} x \leq \int_0^1 f(x)\ \text{d} x + \int_0^1 g(x)\ \text{d} x\;.
\]
Hints:
Let $f,g:[0,1] -> [0,1]$ be continuous functions and $f$ be increasing.
Prove that:
\[
\int_0^1 f(g(x))\ \text{d} x \leq \int_0^1 f(x)\ \text{d} x + \int_0^1 g(x)\ \text{d} x\;.
\]
Hints:
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Consider the function $f(u) - u$, with $u in [0,1]$, and prove that $f(u) - u \leq \int_0^1 f(x) "d" x$; then use Mean Value Theorem to conclude.