Questions tagged [holder-inequality]

Let $a_1$, $a_2$,..., $a_n$, $b_1$, $b_2$,..., $b_n$, $\alpha$ and $\beta$ be positive numbers. Prove that: $$(a_1+a_2+...+a_n)^{\alpha}(b_1+b_2+...+b_n)^{\beta}\geq$$$$\geq\left(\left(a_1^{\alpha}b_1^{\beta}\right)^{\frac{1}{\alpha+\beta}}+\left(a_2^{\alpha}b_2^{\beta}\right)^{\frac{1}{\alpha+\beta}}+...+\left(a_n^{\alpha}b_n^{\beta}\right)^{\frac{1}{\alpha+\beta}}\right)^{\alpha+\beta}$$

Let $a_{ij}>0$ and $\alpha_i>0$. Prove that: $$\prod_{i=1}^k\left(\sum_{j=1}^na_{ij}\right)^{\alpha_i}\geq\left(\sum_{j=1}^n\left(\prod_{i=1}^ka_{ij}^{\alpha_i}\right)^{\frac{1}{\sum\limits_{i=1}^k\alpha_i}}\right)^{\sum\limits_{i=1}^k\alpha_i}$$

Let $f$ and $g$ be positive integrable functions on $[a,b]$ and let $\alpha$ and $\beta$ be positive numbers. Prove that: $$\left(\int\limits_{a}^bf(x)dx\right)^{\alpha}\left(\int\limits_{a}^bg(x)dx\right)^{\beta}\geq\left(\int\limits_{a}^b\left(f(x)^{\alpha}g(x)^{\beta}\right)^{\frac{1}{\alpha+\beta}}dx\right)^{\alpha+\beta}$$