(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.

  • $\begingroup$ 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. $\endgroup$ – reuns Oct 29 '18 at 20:06
  • $\begingroup$ I am aware of the Stirling formula but I don't see why it answers my question. $\endgroup$ – Durac Oct 29 '18 at 20:09
  • $\begingroup$ There is a term $h \log |y|$ appearing in $\log |\Gamma(x+h+iy)|-\log |\Gamma(x+iy)|$ as $y \to \infty$ $\endgroup$ – reuns Oct 29 '18 at 20:15

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