Counterexample of multiplicative Landau inequality in finite interval Question: Is there a sequence of functions $(f_n) \subseteq C^2(I)$  such that
$$\lim_{n\to \infty}\dfrac {\|f_n'\|} {\|f_n\|^{\frac 1 2}\|f_n''\|^{\frac 1 2}} = \infty$$
where $I$ is the unit interval and the norms are all sup-norms?
Landau Inequality states for any $f \in C^2(\mathbb R)$ the following holds
$$\|f'\| \le 2 \|f\|^{\frac 1 2}\|f''\|^{\frac 1 2}.$$
There is a generalization of this inequality for $f \in C^2(\mathbb R^+)$.
For $f \in C^2(I)$, the Gorny's Inequality states
$$\|f'\| \le C \|f\|^{\frac 1 2}\max\{\|f''\|, 2\|f\|\}^{\frac 1 2}.$$
However, the right term of the above inequality is not the same as in Landau Inequality.
I have also found this paper (Landau-Kolmogorov inequality on a finite interval), which says for $f \in C^2(I)$
$$\|f'\|_{[\delta, 1-\delta]} \le C \|f\|^{\frac 1 2}\|f''\|^{\frac 1 2}$$
for some $\delta > 0$.
The Wikipedia page of Landau–Kolmogorov inequality says there are some generalizations of Landau Inequality in finite interval but its expression is very vague, and it provides no references. I wonder whether there is really a constant $C$ that for every $f \in C^2(I)$
$$\|f'\| \le C \|f\|^{\frac 1 2}\|f''\|^{\frac 1 2}.$$
 A: 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.
