Let $\lambda > 0$ and let $f \in L^1(\mathbb{R}) \cap \mathcal{C}^1(\mathbb{R})$ such that $\hat{f} \in L^1(\mathbb{R})$ and such that $\hat{f}$ is zero outside the segment $[-\lambda, \lambda]$. We denote by $r_\lambda$ the function from $\mathbb{R}$ to $\mathbb{C}$ such that $r_\lambda(x) = r(\lambda x)$ for all real $x$. Deduce that, if $f \in L^\infty(\mathbb{R})$, there exists a constant $C \in \mathbb{R}_+^\star$, independent of $\lambda$ and of $f$, such that $$\left\|f'\right\|_\infty \leqslant C\lambda \|f\|_\infty$$
Let $\lambda > 0$ and let $f \in L^1(\mathbb{R}) \cap \mathcal{C}^1(\mathbb{R})$ such that $\hat{f} \in L^1(\mathbb{R})$ and such that $\hat{f}$ is zero outside the segment $[-\lambda, \lambda]$. We denote by $r_\lambda$ the function from $\mathbb{R}$ to $\mathbb{C}$ such that $r_\lambda(x) = r(\lambda x)$ for all real $x$.
Deduce that, if $f \in L^\infty(\mathbb{R})$, there exists a constant $C \in \mathbb{R}_+^\star$, independent of $\lambda$ and of $f$, such that
$$\left\|f'\right\|_\infty \leqslant C\lambda \|f\|_\infty$$