Let $s=\sigma+it$ and $\Gamma(s)$ be the Euler gamma function. Does the inequality holds? $$ \left|\frac{\Gamma(s)}{\Gamma(2-s)}\right|\leq |s|^{2(\sigma-1)},\, 1<\sigma<2,\, t>t_0>0. $$ Difficulties to prove inequality appears when $\sigma$ approximates 1.

Such inequality appeared studying a zeta functions of a second order. Namely, comparing the values of the Selberg zeta-function for the modular subgroup $PSL(2,\mathbb{Z})$ across the critical line: |Z(1-s)|>|Z(s)| (|Z(1-s)|<|Z(s)| ?), $1/2<\sigma<1$.