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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 !

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  • $\begingroup$ I believe this is a special case of en.wikipedia.org/wiki/Gagliardo–Nirenberg_interpolation_inequality ? $\endgroup$ – Calvin Khor Feb 8 at 13:38
  • $\begingroup$ @CalvinKhor I honestly do not know :'( $\endgroup$ – Maman Feb 8 at 15:53
  • $\begingroup$ sorry, I phrased it with a ? but it wasn't a question, if you put $$ (j,m,n,p,q,r,\alpha) = (1,2,1,2,2,2,1/2)$$ then it recovers this result. I can't really name a direct use for it, but one easy consequence is the equivalence of the norm $\|f\|_{L^2} + \|f''\|_{L^2}$ and the usual sobolev norm $ \sqrt{\sum_{i=0}^2 \|f^{(i)}\|_{L^2}^2}$ $\endgroup$ – Calvin Khor Feb 11 at 14:28
  • $\begingroup$ @CalvinKhor I see ! There is just a constant $K$ which is missing with those data ! $\endgroup$ – Maman Feb 11 at 16:38

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