Im trying to prove the positivity of the function $(n+2)\zeta(n+3)-\zeta(n+2)-n-1$ on $\mathbb{N}$ using the inequalities \begin{equation*} \frac{s+1}{s}\zeta(s)\zeta(s+2)\geq \zeta^2(s+1), s>1 \end{equation*} and \begin{equation*} \zeta(s)\zeta(s+2)> \zeta^2(s+1), s>1 \end{equation*} and $$ \zeta^2(s + 1)< \left(\frac{s + 1}{s}\right) \left( \frac{(1 - 2^{-s})(1 - 2^{-(s+2)})}{(1 - 2^{-(s+1)})^2} \right) \zeta(s)\zeta(s + 2), \quad s > 1 $$

I'm trying to prove the positivity of the function $(n+2)\zeta(n+3)-\zeta(n+2)-n-1$ on $\mathbb{N}$ using the inequalities \begin{equation*} \frac{s+1}{s}\zeta(s)\zeta(s+2)\geq \zeta^2(s+1),\quad s>1 \end{equation*} and \begin{equation*} \zeta(s)\zeta(s+2)> \zeta^2(s+1),\quad s>1 \end{equation*} and $$ \zeta^2(s + 1)< \left(\frac{s + 1}{s}\right) \left( \frac{(1 - 2^{-s})(1 - 2^{-(s+2)})}{(1 - 2^{-(s+1)})^2} \right) \zeta(s)\zeta(s + 2), \quad s > 1. $$