My First Inequality :)

27 Sep, 2023

A few days ago, I finally proved my first inequality. Namely,

(a + b + c)^{2} \geq 3 (a b + b c + c a)

This is the first non-trivial inequality that I have formally proved (hopefully more to come) by myself. Sol below.

a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a \geq 3 a b + 3 b c + 3 c a

\sum_{c y c} a^{2} \geq a b + b c + c a

\sum_{c y c} a^{2} + b^{2} \geq 2 a b + 2 b c + 2 c a

\sum_{c y c} a^{2} - 2 a b + b^{2} \geq 0

(a - b)^{2} + (b - c)^{2} + (c - a)^{2} \geq 0

QED.

Anyways this might've been a way for me to test the latex capabilities of Bearblog. It's definitely not the ideal place to use latex, so for now I'll stick to my Substack Blog. Latex support is under consideration so.... maybe I'll return with more math.