I have struggled on this proof. Please a hint, especially related to AM-GM (Cauchy-Schwarz, Jensen are also okay). For \(a,b,c,d \in \mathbb{R}^{+}\), proof that \[\frac{a^2}{b} +\frac{b^2}{c} +\frac{c^2}{d} +\frac{d^2}{a} \geq a+b+c+d\] Thank you very much!

## Comments

Sort by:

TopNewestCheck out Applying AM-GM inequality wiki, and in particular Rearranging creatively. – Calvin Lin Staff · 2 years, 2 months ago

Log in to reply

Adding those, we have \[3(\frac{a^2}{b} + \frac{b^2}{c} +\frac{c^2}{d} +\frac{d^2}{a}) +(a+b+c+d) \geq 4(a+b+c+d)\] \[3(\frac{a^2}{b} + \frac{b^2}{c} +\frac{c^2}{d} +\frac{d^2}{a}) \geq 3(a+b+c+d)\] \[\frac{a^2}{b} + \frac{b^2}{c} +\frac{c^2}{d} +\frac{d^2}{a} ) \geq (a+b+c+d)\] – Figel Ilham · 2 years, 2 months ago

Log in to reply

Could you add this as an example to the wiki page? – Calvin Lin Staff · 2 years, 2 months ago

Log in to reply

– Figel Ilham · 2 years, 2 months ago

I'll tryLog in to reply

– Figel Ilham · 2 years, 2 months ago

I'll try firstLog in to reply