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Fie \( f:[0,1]\to [0, \infty) \) o functie integrabila. Sa se arate ca

\( \int_0^1 f(x)dx\cdot\int_0^1f^{2}(x)dx\leq\int_0^1f^{3}(x)dx \).

Laurentiu Panaitopol, locala Bucuresti 2005

Folosim CBS de doua ori:

\( \int_{0}^{1}f^3(x)dx\int_{0}^{1}f(x)dx=\int_{0}^{1}(f(x)sqrt{f(x)})^2dx \int_{0}^{1} (sqrt{f(x)})^2dx\geq(\int_{0}^{1}f^2(x)dx)^2= \)
\( \int_{0}^{1}f^2(x)dx\int_{0}^{1}f^2(x)dx\int_{0}^{1}dx\geq \int_{0}^{1}f^2(x)dx(\int_{0}^{1} f(x)dx)^2. \)

Daca \( \int_{0}^{1} f(x)dx>0 \), atunci rezulta inegalitatea.
Daca \( \int_{0}^{1} f(x)dx=0 \), atunci rezulta \( \int_{0}^{1} f^3(x)dx\geq 0 \).