Hlder's inequality
The Hölder's Inequality, a generalization of the Cauchy-Schwarz inequality, states that,
For all such that $\frac {1}{p} \plus{} \frac {1}{q} \equal{} 1,$ (Error compiling LaTeX. Unknown error_msg) we have:
$\sum_{i \equal{} 1}^n a_ib_i\leq \left(\sum_{i \equal{} 1}^n a_i^p\right)^{\frac {1}{p}}\left(\sum _{i \equal{} 1}^n b_i^q\right)^{\frac {1}{q}}.$ (Error compiling LaTeX. Unknown error_msg)
Letting in this inequality leads to the Cauchy-Schwarz Inequality.
This can also be generalized further to sets of variables with a similar form.
Applications
1. Given we have,
2. Power-mean inequality: For and we have