FKMO 1997 Problem 5

For positive numbers $a_1,a_2,\dots,a_n$ , we define $$A=\frac{a_1+a_2+\cdots+a_n}{n}, G=\sqrt[n]{a_1\cdots a_n}, H=\frac{n}{a_1^{-1}+\cdots+a_n^{-1}}$$ Prove that

(i) $\frac{A}{H}\leq -1+2\left(\frac{A}{G}\right)^n$ , for n even

(ii) $\frac{A}{H}\leq -\frac{n-2}{n}+\frac{2(n-1)}{n}\left(\frac{A}{G}\right)^n$ , for $n$ odd