Inegalitatea lui Karamata
Posted: Sun Feb 01, 2009 1:00 pm
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 \( k=\frac{\sqrt{m_{1}m_{2}}+\sqrt{M_{1}M_{2}}}{\sqrt{m_{1}M_{2}}+\sqrt{M_{1}m_{2}}} \).
\( k^{-2}\leq\displaystyle\frac{\int_0^1f(x)dx\cdot\int_0^1g(x)dx}{\int_0^1f(x)g(x)dx}\leq k^2, \)
unde \( k=\frac{\sqrt{m_{1}m_{2}}+\sqrt{M_{1}M_{2}}}{\sqrt{m_{1}M_{2}}+\sqrt{M_{1}m_{2}}} \).