$f \in C^2(0,\infty)$ , then how to show that $\Big(\sup_{x>0} |f'(x)|\Big)^2\le 4\big(\sup _{x>0}|f(x)|\big)\big(\sup_{x>0}|f''(x)|\big)$ ?
Prove $\sup \left| f'\left( x\right) \right| ^{2}\leqslant 4\sup \left| f\left( x\right) \right| \sup \left| f''\left( x\right) \right| $