# Prove with holder $\frac{a_1^2}{a_2}+\frac{a_2^2}{a_3}+\dots+\frac{a_n^2}{a_1}\ge a_1+a_2+\dots a_n$

Prove the following inequalities with Holder.

1. Prove that for all positive real numbers $$a,b,c,x,y,z$$ $$\frac{a^3}{x}+\frac{b^3}{y}+\frac{c^3}{z}\ge \frac{(a+b+c)^3}{3(x+y+z)}$$

2. Prove that $$\frac{a_1^2}{a_2}+\frac{a_2^2}{a_3}+\dots+\frac{a_n^2}{a_1}\ge a_1+a_2+\dots a_n.$$

For the second problem. I think Titu engel form suffices.

We get $$\frac{a_1^2}{a_2}+\frac{a_2^2}{a_3}+\dots+\frac{a_n^2}{a_1}\ge \frac{(a_1+a_2\dots+a_n)^2}{a_1+a_2+\dots a_n}=a_1+a_2+\dots a_n.$$

Note sure about the Holder proof though. Any solutions?

• What is $\frac{a_2^2}{+}$?
– Gary
Oct 14, 2021 at 5:00
• @Gary I think he meant $a_3$ .. It's a pretty well known ineq., Oct 14, 2021 at 5:16
• @SunainaPati Then please fix it in the title and the last displayed equation too.
– Gary
Oct 14, 2021 at 5:24
• For those who don't know Titu-Engel-Sedrakian inequality see here Oct 14, 2021 at 5:59

1. This is Holder's in the form $$\left(\sum\frac{a^3}x\right)\left(\sum x\right)\left(\sum1\right)\ge\left(\sum a\right)^3.$$