Question: Let the function $\displaystyle f$ be holomorphic in the open set $\displaystyle G$. Prove that the function $\displaystyle g(z) = \bar{f\left( \bar{z}\right)}$ is holomorphic in the set $\displaystyle G^\ast =\{\bar{z}: z \in G\}$

Side note: the first bar is over the z and the second bar is over the image of f(z)

My idea for the proof:

I want to show that there are first partial derivatives in G* s.t. the partial derivatives are continuous and satisfy the Cauchy Reimann equations

$\displaystyle du \setminus dx=dv\setminus dy $ $\displaystyle du \setminus dy=-dv \setminus dx$

I differentiate this function on G* w.r.t. to x and y to derive the first partial derivatives.

(how i do this is f(x+h,y)-f(x,y) over h and f(x,y+h) -f(x,y) over ih)

However when I do this I get df/dx = du/dx + -i dv/dx

and df/dy = -dv/dy - i du/dy

which implies the function is constant.

Can someone explain to me what I am doing wrong or not seeing?