I would be interested in proving the following inequality involving Dirichlet's eta function $\eta(s)$ at different values which, after some numerical investigations, I am sure is true
$|\chi(1-s)\eta(s+2)-\eta(3-s)|>0$ $\qquad$ $1/2<\Re(s)<1$
Here $\chi(1-s)=\frac{\Gamma(s)}{(2\pi)^s}2\cos\left(\frac{\pi s}{2} \right)\frac{1-2^s}{1-2^{1-s}}$ is the factor entering in the functional equation of the Dirichlet eta function
$\eta(1-s)=\chi(1-s)\eta(s)$
Do you have any suggestion on how to attack this problem and/or references that could be helpful? Thank you very much in advance!