Is it true that $2^p\|f\|^p_{L_p}+2^p\|f'\|^p_{L_p} >2\|f\|^p_{L_\infty}$?

Question

For any $C^1$ function defined in $(0,1)$, is it true that $$ 2^p\|f\|^p_{L_p}+2^p\|f'\|^p_{L_p} >2\|f\|^p_{L_\infty} $$ If it is true, how to prove it?

Answer 1

If $x,y \in (0,1)$ then $$|f(x)| \le |f(y)| + |f(x) - f(y)| \le |f(y)| + \int_x^y |f'(t)| \, dt$$ so that $$ |f(x)| \le |f(y)| + |x-y|^{1-1/p} \left( \int_x^y |f'(t)|^p \, dt\right)^{1/p}$$ by e.g. Holder's inequality. This leads to $$|f(x)|^p \le 2^p |f(y)|^p + 2^p |x-y|^{p-1} \int_x^y |f'(t)|^p \, dt \le 2^p |f(y)|^p + 2^p \int_0^1 |f'(t)|^p \, dt.$$

You could now integrate with respect to $dy$.