Question: Let the function be holomorphic in the open set . Prove that the function is holomorphic in the set

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

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?