I tried differentiating it and get a whole mess:
$$f'(p)=(u^p+1)^{1/p}\left(\frac{u^p\log{u}}{p(u^p+1)}-\frac{\log{(u^p+1)}}{p^2}\right)$$
And I don't know how to prove that this is always negative when $u>0$ and $p>1$.
I tried differentiating it and get a whole mess:
$$f'(p)=(u^p+1)^{1/p}\left(\frac{u^p\log{u}}{p(u^p+1)}-\frac{\log{(u^p+1)}}{p^2}\right)$$
And I don't know how to prove that this is always negative when $u>0$ and $p>1$.
(I trust your differentiation.) In the parentheses, it is
$$\frac{u^p\log u^p - (u^p + 1)\log(u^p + 1)}{p(u^p + 1)}.$$
Just look at the numerator.