I am Reading this work on page 21. In the said page, the author reach the inequality: $$\int|\zeta\nabla u^{\beta/p}|^p\leq\Big(\frac{\beta}{\beta-p-1}\Big)^p\int|u^{\beta/p}\nabla\zeta|^p$$
By using the fact that $$|\nabla(\zeta u^{\beta/p})|\leq |\zeta\nabla u^{\beta/p}|+|u^{\beta/p}\nabla\zeta|$$
and the Minkpwski inequality he conclude that $$\int_\Omega |\nabla(\zeta u^{\beta/p})|^p\leq\Big(\frac{2\beta-p+1}{\beta-p+1}\Big)^p\int_\Omega|u^{\beta/p}\nabla\zeta^p|$$
I am having trouble with the last inequality. Im my calculations it goes in the reverse direction, to wit: $$\Big(\frac{\beta}{\beta-p+1}+1\Big)^p\leq2^{p-1}\Big(\frac{\beta}{\beta-p+1}\Big)^p+2^{p-1}$$
Any light on it is appreciated. Thank You
