Let $A\subseteq C$ be an open connected domain and $f$ a holomorphic in $A$. I want to show that: if $f(A)$ is a subset of a line on the complex plane, then $f$ is constant.
I began with expressing every $z_0$ on some line $\gamma (t) =at+b$ on the complex plane as: $z_0=x_0+i[ax_0+b]$
So: $f(z)=x+i[ax+b]$ , but I'm stack (pun intended) with one variable. I doubt that this syllogism is correct. Any help on that one?
Suppose first that the line in question is the real line. If you write $f(x+yi)$ as $u(x,y)+v(x,y)i$, then this means that $v$ is the null function. By the Cauchy-Riemann equations, $u_x=v_y=0$ and $u_y=-v_x=0$. Therefore, $u$ is constant and so $f$ is constant.
In the general case, take complex numbers $\omega$ and $\tau$ such that $\omega\neq0$ and that the image of $\omega f+\tau$ is a subset of $\mathbb R$. Then $\omega f+\tau$ is constant and therefore $f$ is constant.