For a fixed value of $a<\frac{1}{4}$ prove that
$\ |x|^{2a} < c_1(a) \frac{1+|x|}{1+|x|^{1-2a}}$
holds for all $x\in\mathbb R$
Similarly show that for fixed $a<\frac{1}{2}$
$\ |x|^{2a} < c_2(a) \frac{1+x^2}{1+|x|^{2-2a}}$
holds for all $x\in\mathbb R$
"i have encountered this inequality in existence result of fully nonlinear evolutionary Navier Stokes Equations..Exactly at : Navier Stokes Equations Theory and Numerical Analysis by Roger Temam pages;277,286. Thanks for your interest indeed." ($c_1(a)$ and $c_2(a)$ are constants depending on a)