Let $f\in \mathcal{C}^2(\mathbb{R},\mathbb{R})$ and suppose that $\int_\mathbb{R}f^2<+\infty$ and $\int_\mathbb{R}f''^2<+\infty$. Then we can deduce that : $\left(\int_{\mathbb{R}}f'^2\right)^2 \le (\int_\mathbb{R}f^2)(\int_\mathbb{R}f''^2)$.
In the standard proof, we use the famous inequality of Cauchy-Schwarz. However, I was wondering if there was a well-known mathematician who proved that statement and also for what kind of applications we can use it ? If someone has references it would be a nice idea to share.
PS : there is a similar inequality called Landau-Kolmogorov.
Thanks in advance !