Let $s=\sigma+it$ and $\Gamma(s)$ be the Euler gamma function. Does the inequality hold? $$ \left|\frac{\Gamma(s)}{\Gamma(2-s)}\right|\leq |s|^{2(\sigma-1)},\, 1<\sigma<2,\, t\in \mathbb{R}. $$ 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)| ?), $1/2<\sigma<1$.

We can show that $$ \left|\frac{\Gamma(s)}{\Gamma(2-s)}\right|\leq \left|s-2-\frac{\sqrt{2}}{2}\right|^{2(\sigma-1)},\, 1<\sigma<2,\, t\in \mathbb{R}. $$ See the Lemma 7 in http://link.springer.com/article/10.1007%2Fs00025-015-0486-7#page-1

However, the same technic doesn't work for the first inequality.