1999 Putnam B4

Let $f$ be a real function with a continuous third derivative such that $f(x),

f'(x), f''(x), f'''(x)$are positive for all$x$. Suppose that

$f'''(x)\leq f(x)$ for all $x$. Show that $f'(x)<2f(x)$ for all $x$.

令 $f$ 为具有连续三阶导数的实函数，使得 $f(x)，

f'(x)、f''(x)、f'''(x)$对于所有$x$ 均为正。假设

$f'''(x)\leq f(x)$ 对于所有 $x$。证明对于所有 $x$，$f'(x)<2f(x)$。