A common, and versatile inequality.

1𝑛𝑛𝑖=1𝑥𝑖=𝑛𝑥1𝑥2𝑥𝑛

Trivial extension:

1𝛼𝛼𝑖𝑥𝑖=𝛼𝑥𝑎𝑖𝑖

Where 𝛼=𝛼𝑖.

Cauchy-Schwarz inequality

Applications

minimum sum of distances to asymptotes
minimum distance from point to improper rational function

References