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

Question

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?

Answer 1

1. This is Holder's in the form $$\left(\sum\frac{a^3}x\right)\left(\sum x\right)\left(\sum1\right)\ge\left(\sum a\right)^3.$$
2. Titu is just a form of Cauchy Schwarz, which itself is a special case of Holder.