Show that holomorphic function $f: \mathbb{C} \rightarrow \mathbb{C}$ is constant Let's $f: \mathbb{C} \rightarrow \mathbb{C}$ be a holomorphic function such that values $f$ are on line $y=ax+b$. Show that $f$ is constant.
I think I should use Cauchy-Riemann equations but I don't know what does mean that values $f$ are on line $y=ax+b$. Can you explain me that?
Thanks in advance.

Your hypothesis reads: $$f(x+iy) = u(x,y) + i(a u(x,y) + b),$$ that is, the point $f(x+iy)$ as a point in $\Bbb R^2$ belongs to the said line. The Cauchy-Riemann equations read, abbreviating the notation: $$u_x = a u_y, \quad u_y = -au_x.$$ So $u_x = -a^2u_x \implies (1+a^2)u_x = 0$ and... oh, you can conclude now! (if you need one more push, please tell me)
