Problem 936 (difficulty: 3/10)

Prove the following

(a) If \(\displaystyle f,g:[a,b]\to\R\) are integrable, then \(\displaystyle \left(\int_a^b fg\right)^2 \le \left(\int_a^b f^2\right) \left(\int_a^b g^2\right) \) (Schwarz-inequality).

(b) If \(\displaystyle f,g:[a,b]\to\R\) are integrable and \(\displaystyle p,q>0\) such that \(\displaystyle \frac1p+\frac1q=1\), then \(\displaystyle \int_a^b fg \le \left(\int_a^b |f|^p\right)^{1/p} \left(\int_a^b |g|^q\right)^{1/q} \) (Hölder-inequality).

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