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?

  • 1
    $\begingroup$ What is $\frac{a_2^2}{+}$? $\endgroup$
    – Gary
    Oct 14, 2021 at 5:00
  • $\begingroup$ @Gary I think he meant $a_3$ .. It's a pretty well known ineq., $\endgroup$ Oct 14, 2021 at 5:16
  • $\begingroup$ @SunainaPati Then please fix it in the title and the last displayed equation too. $\endgroup$
    – Gary
    Oct 14, 2021 at 5:24
  • $\begingroup$ For those who don't know Titu-Engel-Sedrakian inequality see here $\endgroup$
    – Jean Marie
    Oct 14, 2021 at 5:59

1 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.

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