Inequality of $L^p \left ( \mathbb{R} \right )$ norms of functions and their derivatives

Question

I'm trying to prove that there is a constant $\space C>0 \space$ such that for any function $\space u \in C^{\infty}_{0} \left ( \mathbb{R} \right ) \space$ following inequality holds:

${\left \| u \right \|}_{L^2 \left ( \mathbb{R} \right )} \leq C {\left \| u' \right \|}^{1/2}_{L^1 \left ( \mathbb{R} \right )} {\left \| u \right \|}^{1/2}_{L^1 \left ( \mathbb{R} \right )}$

I'm actually trying to get much stronger result, but I don't even know how to prove this one.

I was thinking about using the fundamental theorem of calculus and Hölder's inequality, but I don't know how exactly.

I'll appreciate any help.

Answer 1

This is a particular case of the Gagliardo-Niremberg inequality with (notation as in the Wikipedia article) $$ n=1,\quad j=0,\quad p=2,\quad r=q=1,\quad \alpha=\frac12. $$