In our recent researches, we obtain the following lower bound for the (real) Riemann zeta function: \begin{align*} \zeta(s)\geq \frac{5 - 2s}{s - 1};\;\;\;\; 1<s<\frac{5}{2}. \end{align*} (a) Is it a known result?
(b) What about the case $s>\frac{5}{2}$?
(c) Do you know any useful references?
Note. We know that $\zeta(s)\geq\frac{1}{s -1}$ for $s>1$ ( because, we have $\frac{1}{k^s}\geq\frac{1}{x^s}$ for every $k\in\mathbb{N}$ and $x\geq k$, where $s$ is constant. So, we get the result). Therefore, the lower bound $\frac{5 - 2s}{s - 1}$ is stronger than $\frac{1}{s - 1}$ if $1<s<2$.
This cannot be a known result because it is wrong: For $s=3/2$ you have
$$\zeta(s) \approx 2.6123753486854883, \quad \frac{5-2s}{s-1}=4$$