The derivative of the Gamma function

Question

The Gamma function satisfies the relation $z\Gamma(z)=\Gamma(z+1)$, whence $|\Gamma(z+1)|>|\Gamma(z)|$ whenever $|z|>1$ (and $z$ is not a non-positive integer). This naturally leads us to the idea that for a fixed $y>1$ the function $|\Gamma(x+iy)|$ of the variable $x$ could be increasing. Is this true? I would much appreciate a reference or a proof. The area $\{x+iy: \ x>1, \ y<1\}$ is of interest as well.

Answer 1

This is wrong. Here a plot for $|\Gamma(x+1.01 \cdot i)|$

$$|\Gamma(0.2+1.01\cdot i)|= 0.516403048428670750862809525289$$ $$|\Gamma(0.6+1.01\cdot i)|= 0.510624247985962860913065928364$$