Question 93
Let $a, b$ be two nonnegative real numbers and $n$ a positive integer. Prove that
$$
\left(1-2^{-n}\right)\left|a^{2^{n}}-b^{2^{n}}\right| \geq \sqrt{a b}\left|a^{2^{n}-1}-b^{2^{n}-1}\right|
$$
$$
\left(1-2^{-n}\right)\left|a^{2^{n}}-b^{2^{n}}\right| \geq \sqrt{a b}\left|a^{2^{n}-1}-b^{2^{n}-1}\right|
$$