# The derivative of the Gamma function, once more

(In connection with this. My previous question was answered, and here is a modified version of my 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). Is the function $$|\Gamma(x+iy)|$$ of the variable $$x$$ increasing if the parameter $$y$$ is sufficiently large? I am also interested in a description of the area where (the local version of) this property is fulfilled.

• Are you aware of the explicit formula for $\log \Gamma$ and the complex Stirling approximation and why it is an answer ? That said, it could follow from a more elementary argument. – reuns Oct 29 '18 at 20:06
• I am aware of the Stirling formula but I don't see why it answers my question. – Durac Oct 29 '18 at 20:09
• There is a term $h \log |y|$ appearing in $\log |\Gamma(x+h+iy)|-\log |\Gamma(x+iy)|$ as $y \to \infty$ – reuns Oct 29 '18 at 20:15