Complex Analytic functions that are lines are constant If a complex function is analytic on D and its point lie on a line, it is constant with respect to x and y. How does one prove this? 

Here is an elementary proof: There exist $m,c \in \mathbb R$ such that $v=mu+c$ where $u$ and $v$ are the real and imaginary parts of $f$. This gives $v_x=mu_x$ and $v_y=mc_y$. Use Cauchy Riemann equations to get $(1+m^{2})u_x=0$. Hence $u_x=0$. Similarly $u_y=0$ so $u$ is a constant. Similarly $v$ is also a constant and so is $f$. [Notation: $u_x=\frac {\partial } {\partial x} u$ etc]. 
