The analytic function $f:\mathbb{C}\rightarrow\mathbb{C}$, $\ f(z)=u+iv \ $ with the property that $$|f(z)|=\varphi\left(\frac{x^2+y^2}{x}\right)$$
where $\varphi$ is a function depending on $\frac{x^2+y^2}{x}$
Any hint to solve this? I don't need a complete solution.
Thanks!
I think what you're saying is that the curves $|f(z)|=constant$ have the form $\dfrac{x^2 + y^2}{x} = constant$ (at least for $x \ne 0$: I don't know what you want to happen for $x=0$). Such curves (as you see from multiplying by $x$ and completing the square) are circles passing through the origin. But there are problems with that: $|f(0)|$ can only be one number. Thus the answer to the question as stated is that $f$ must be constant: any nonconstant $f$ couldn't be analytic at $0$.
On the other hand, if $f$ is allowed to be analytic on $\mathbb C \backslash \{0\}$ you can have solutions
$f(z) = e^{c/z}$ for real constants $c$.