Prove the following inequalities with Holder.
Prove that for all positive real numbers $a,b,c,x,y,z$ $$\frac{a^3}{x}+\frac{b^3}{y}+\frac{c^3}{z}\ge \frac{(a+b+c)^3}{3(x+y+z)} $$
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?