Consider the following function :
$$f(x)=\frac{x\Gamma'(x)}{(\Gamma(x))^2}-\frac{1}{\Gamma(x)}$$
I need to calculate the following limit:
$$\lim_{y\rightarrow\infty}\frac {|f(x+iy)|}{e^{2πy}}$$
I 'guessed' with some approximate analysis that this limit is zero but I couldn't be sure of that.
For that guess I used the estimate :
$$|\Gamma\left(x+iy\right)|\sim\sqrt{2\pi}|y|^{x-(1/2)}e^{-\pi|y|/2} (1+O(\frac{1}{|y|}))$$
As $|y|\rightarrow\infty$
$C$ and $c$ are constants
And so, $$\lim_{y\rightarrow\infty}\frac {f(x+iy)}{e^{2πy}}= 0$$