I'm reading a demonstration that uses this following inequality. For a fixed $\gamma<1/4$, exists a $c_5(\gamma)$, such that
$|\tau|^{2\gamma} \leq c_5(\gamma) \frac{1+|\tau|}{1+|\tau|^{1-2\gamma}}, \forall \tau \in \mathbb{R}$.
I tried to deduce that using the fact that $0<\frac{1}{1+|\tau|}\leq1$, but i couldn't get in anywhere.