I found the following statement in a textbook:
$$\int\mathrm{d}^{4}x \vert f(x)\vert^{2}=\int\mathrm{d}^{4}x (C+\vert x\vert^{8})\cdot (C+\vert x\vert^{8})^{-1}\vert f(x)\vert^{2}\leq\mathrm{sup}_{y\in\mathbb{R}^{4}}[(C+\vert y\vert^{8})\vert f(y)\vert^{2}]\int\mathrm{d}^{4}x (C+\vert x\vert^{8})^{-1}$$
First of all: Why am I allowed to use this inequality...Is this a standard estimation of integral? If yes, has it name? Secondly, why uses the author $\vert x\vert^{8}$ and not an other power? Is it because with the 8th power the integral exists?