I have a function of complex variable $z=x+iy$, $f(z)$, which is analytic in the domain $\{z : \Re(z)>0 \}$, $f(iy)=f(-iy)$ and $f(z)=O(|z|^{-3/2})$ when $|z|\to +\infty$.
Moreover, if $z$ has zero imaginary part, then $f(x)=O(|x|^{-3/2})$, when $|x| \to +\infty$.
I have to study whether $f(z)/(1+z^2 f(z))$ is summable, so i'm interested in the behavior of the latter function's modulus when $|z|>>1$. I'm not sure about the accuracy of my method; all suggestions are welcome!
Since $\quad |1+z^2 f(z)|=\sqrt{1+|z^2f(z)|^2 + 2\Re(z^2f(z))}$,
and when $|z|>>1$,
$|z^2 f(z)|^2=|z|^4 |f(z)|^2=|z|^4 O(|z|^{-3/2})^2=O(|z|) \qquad \text{and }\qquad \Re(z^2f(z))=O(|\Re(z)|^{1/2})$,
I get: $\qquad |1+z^2 f(z)|=\sqrt{1+O(|z|) + 2O(|\Re(z)|^{1/2}})$.
Then I would say that
$\frac{|f(z)|}{|1+z^2 f(z)|}=O(|z|^{-2})$, when $|z|>>1$.