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Found problems: 1

1985 Traian Lălescu, 2.2
Tags: inequalities , calculus , young s inequality , integral inequality , bijective , integral , real analysis
Let $ a,b,c\in\mathbb{R}_+^*, $ and $ f:[0,a]\longrightarrow [0,b] $ bijective and non-decreasing. Prove that: $$ \frac{1}{b}\int_0^a f^2 (x)dx +\frac{1}{a}\int_0^b \left( f^{-1} (x)\right)^2dx\le ab. $$