Is it true that,$ \|f\|^p_{L_p} \le 2^{p-1}\|u'\|^p_{L_p}+2^{p-1}|u(\zeta)|^p $?

Question

Is it true that,$$ \|f\|^p_{L_p} \le 2^{p-1}\|f'\|^p_{L_p}+2^{p-1}|f(\zeta)|^p $$

for $f$ is $C^1$ in $[0,1]$ and $\zeta$ in [0,1]

Answer 1

You can write $f(x)= f(\xi)+\int_{\xi}^xf'(s)ds$ for $0\leq \xi<x\leq 1$ and then you can proceed. Also you will need the elementary inequality $(a+b)^p\leq 2^{p-1}(a^2+b^2)$.