Proof that $h^{2H} \cdot \frac12 \left[ (n+1)^{2H} - 2n^{2H} + (n-1)^{ 2H} \right]$ has only one root in $[0,1$ for all $n\in \mathbb{N}$

Question

So, I am a little bit confused, how can I proof that $$f(H):=h^{2H} \cdot \frac12 \left[ (n+1)^{2H} - 2n^{2H} + (n-1)^{ 2H} \right]$$ has only one root in $[0,1]$ for all $h>0$ and $n\in\mathbb{N}$?

It's easy to see the root $H=\frac12$.