One variable inequality with parameter

Using the power-mean inequality, one can prove that for all $p\geq 1$ and for all $x\geq 0$ we have that $$(x^{p+1}+1)^2\geq \left(\frac{x^2+1}{2}\right)^p(x+1)^2.$$The wolfram suggests that this inequality holds also for $p\in [0,1).$ Is there any proof for this?