I have the following question
Suppose that $h(x,y)$, with no zeros in $\mathbb{D}$ with harmonic with conjugate harmonic conjugate $g(x,y)$. Show $f(x,y) = \log(h(x,y)^2 + g(x,y)^2)$ is harmonic in $\mathbb{D}$.
The question says that you are not meant to calculate a bunch of partial derivatives. Thus, I think that you are suppose to express $f$ has a real or imaginary of an analytic function in $\mathbb{D}$. We can write
$$k(x + iy) = h(x,y) + i v(x,y)$$
where $k$ is analytic (and nonzero) in $\mathbb{D}$. Since $k$ is analytic in $\mathbb{D}$, nonzero, and $\mathbb{D}$ is simply connected, we can define a branch of $\log(k^2)$. I am wondering if I can say that $\log(k^2) = \log|k| + i \arg(k^2)$ and then conclude that $f$ is harmonic since $\log|k| = f(x,y)$. I am not sure if I am allowed to express $\log(k^2)$ is this form.
Any comments or suggestions would be greatly appreciated.