Question 5
Let $a,s$ be real numbers with $s\gt 1$, and let $f:(0,+\infty)\to\mathbb R$. (1) Suppose $a\gt 1$. Prove that if (i) $f(x)^2\le ax^s f(x/a)$ for every $x\gt 0$, and (ii) $f(x)\lt 2^{2000}$ for every $0\lt x\lt 2^{-2000}$, then $f(x)\le x^s a^{1-s}$ for every $x\gt 0$. (2) Suppose $a\gt 2$. Find a function satisfying (i) for which $f(x)\gt x^s a^{1-s}$ for every $x\gt 0$.