You can successively establish the following:

$$\| f \|_2\le \| \|_\infty,$$

$$\|\hat f\|_2\le C\|f\_2,$$

$$\|f'\|_\infty\le C\gamma^{3/2}\|\hat f\|_2.$$

Here $\|\cdot\|_p$ denotes the $L^p$-norm. The last inequality uses the assumption about the support of $\hat f$.

You can successively establish the following:

$$\| f \|_2\le C\sup (1+|y|)|f(y)|,$$

$$\|\hat f\|_2\le C\|f\|_2,$$

$$\|f'\|_\infty\le C\gamma^{3/2}\|\hat f\|_2.$$

Here $\|\cdot\|_p$ denotes the $L^p$-norm. The last inequality uses the assumption about the support of $\hat f$.

You can successively establish the following:

$$\|f\|_2 \le C \|(1+|y|)f\|_\infty,$$

$$\|\hat f\|_2\le C\|f\|_2,$$

$$\|f'\|_\infty\le C\gamma^{3/2}\|\hat f\|_2.$$

Here $\|\cdot\|_p$ denotes the $L^p$-norm. The last inequality uses the assumption about the support of $\hat f$.

You can successively establish the following:

$$\| f \|_2\le \| \|_\infty,$$

$$\|\hat f\|_2\le C\|f\_2,$$

$$\|f'\|_\infty\le C\gamma^{3/2}\|\hat f\|_2.$$

Here $\|\cdot\|_p$ denotes the $L^p$-norm. The last inequality uses the assumption about the support of $\hat f$.