Inegalitatea lui Karamata

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Fie \( f, g \) doua functii integrabile. Daca \( 0<m_{1}\leq f(x)\leq M_{1} \) si \( 0<m_{2}\leq g(x)\leq M_{2} \), atunci este adevarata inegalitatea

\( k^{-2}\leq\displaystyle\frac{\int_0^1f(x)dx\cdot\int_0^1g(x)dx}{\int_0^1f(x)g(x)dx}\leq k^2, \)

unde \( k=\frac{\sqrt{m_{1}m_{2}}+\sqrt{M_{1}M_{2}}}{\sqrt{m_{1}M_{2}}+\sqrt{M_{1}m_{2}}} \).