Putnam 1999, Problem B-4

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 \(x\).

Solution:
(based on work by Daniel Stronger) We make repeated use of the following fact: if $f$ is a differentiable function on all of $\mathbb{R}$, $\lim_{x \to -\infty} f(x) \geq 0$, and $f'(x) > 0$ for all $x \in \mathbb{R}$, then $f(x) > 0$ for all $x \in \mathbb{R}$. (Proof: if $f(y) < 0$ for some $x$, then $f(x) < f(y)$ for all $x < y$ since $f' > 0$, but then $\lim_{x \to -\infty} f(x) \leq f(y) < 0$.)

From the inequality $f'''(x) \leq f(x)$ we obtain
$$f''f'''(x) \leq f''(x)f(x) < f''(x)f(x) + f'(x)^2$$
since $f'(x)$ is positive. Applying the fact to the difference between the right and left sides, we get
$$\frac{1}{2}(f''(x))^2 < f(x)f'(x). \tag{1}$$

On the other hand, since $f(x)$ and $f'''(x)$ are both positive for all $x$, we have
$$2f'(x)f''(x) < 2f'(x)f''(x) + 2f(x)f'''(x).$$

Applying the fact to the difference between the sides yields
$$f'(x)^2 \leq 2f(x)f''(x). \tag{2}$$

Combining (1) and (2), we obtain
$$\frac{1}{2}\left(\frac{f'(x)^2}{2f(x)}\right)^2 < \frac{1}{2}(f''(x))^2$$
$$< f(x)f'(x),$$
or $(f'(x))^3 < 8f(x)^3$. We conclude $f'(x) < 2f(x)$, as desired.

Note: one can actually prove the result with a smaller constant in place of 2, as follows. Adding $\frac{1}{2}f'(x)f'''(x)$ to both sides of (1) and again invoking the original bound $f'''(x) \leq f(x)$, we get
$$\frac{1}{2}[f'(x)f'''(x) + (f''(x))^2] < f(x)f'(x) + \frac{1}{2}f'(x)f'''(x)$$
$$\leq \frac{3}{2}f(x)f'(x).$$

Applying the fact again, we get
$$\frac{1}{2}f'(x)f''(x) < \frac{3}{4}f(x)^2.$$

Multiplying both sides by $f'(x)$ and applying the fact once more, we get
$$\frac{1}{6}(f'(x))^3 < \frac{1}{4}f(x)^3.$$

From this we deduce $f'(x) < (3/2)^{1/3}f(x) < 2f(x)$, as desired.

I don't know what the best constant is, except that it is not less than 1 (because $f(x) = e^x$ satisfies the given conditions).