Counterexample of multiplicative Landau inequality in finite interval

The ratio is not defined if $\Vert f_n''\Vert = 0$, i.e. for linear functions. But even if we restrict the problem to nonlinear functions, the ratio can be arbitrarily large.

An example: $f_n(x) = x^2 + nx $ for $x \in [0, 1]$. Then $$ \frac{\Vert f_n' \Vert}{\Vert f_n \Vert^{1/2} \cdot \Vert f_n'' \Vert^{1/2}} = \frac{2+n}{\sqrt{1+n} \cdot \sqrt 2} \to \infty $$ for $n \to \infty$.

Remark: There is a related result from Landau in

Landau, E. (1913). "Ungleichungen für zweimal differenzierbare Funktionen". Proc. London Math. Soc. 13: 43–49.

where it is shown (“Satz 1”):

Theorem: If $f$ is twice differentiable on an interval $I$ of length $\ge 2$ with $|f(x)| \le 1$ and $|f''(x)| \le 1$ on $I$, then $|f'(x)| \le 2$ on $I$.

By scaling the argument, Landau obtains the following corollary for $a > 0$, $b > 0$:

If $g$ is twice differentiable on an interval $I$ of length $\ge 2 \sqrt{a/b}$ with $|g(x)| \le a$ and $|g''(x)| \le b$ on $I$, then $|g'(x)| \le 2\sqrt{ab}$ on $I$.

But due to the given restrictions, this does not lead to an upper bound for $\frac{\Vert f' \Vert}{\Vert f \Vert^{1/2} \cdot \Vert f'' \Vert^{1/2}}$ for arbitrary functions on arbitrary intervals of finite length.