$f$ be an entire function whose value lie in a straight line in the complex plane I was thinking about the problem that says:
Let $f$ be an entire function whose values lie in a straight line in the complex plane. Then which of the following option(s) is/are correct?

(a) $f$ is necessarily identically equal to zero,
(b) $f$ is constant,
(c) $f$ is Möbius map,
(d) $f$ is a linear function.

From the fact that the values of f lie in a straight line,we can take f(x) to be of the form ax+b; a,b being constants. But in this case ,we can not conclude that f is a linear function as the term linear function refers to a function that satisfies the following two properties:
f(x+y)=f(x)+f(y) and f(ax)=af(x). But we see that f does not satisfy the very first property. So option (d) does not hold good.We can also eliminate (a),(b) in the process.So i think $f$ represents a Möbius map.Does my thinking go in the right direction?
Please help. Thanks in advance for your time.
 A: You can use the Open Mapping Theorem, which states:
Any non-constant holomorphic function that is defined on a non-empty open subset $ U $ of $ \mathbb{C} $ must map open subsets of $ U $ to open subsets of $ \mathbb{C} $.
As $ f[\mathbb{C}] $ is forced to be contained in a straight line, and as a straight line cannot contain a non-empty open subset of $ \mathbb{C} $, it follows that $ f $ must be a constant function. The correct answer is therefore (b).
A: This is really an extended comment.  You seem to have a lot of misconceptions about linear functions.  Let me clarify some things:
(1) As I said in the comments: the situation you have is a complex-valued function of a complex variable $w = f(z)$ whose image is a line.  You cannot conclude from this alone that $f(z) = az + b$ for some constants $a, b$.
I think you are getting confused with the case of a real-valued function of a real variable $y = f(x)$ whose graph is a line.  In this case you can say that $f(x) = ax + b$.
There are two distinctions here.  First, there is a difference between real functions of a real variable and complex functions of a complex variable.  Second, there is a difference between "image" and "graph."
(2) You have to be careful when you say "linear function."
In many calculus classes -- and unfortunately also in some analysis classes -- the term "linear function" means a function of the form $f(x) = ax + b$.
However, this is not the best terminology.  A function of the form $f(x) = ax + b$ should be called an affine function.  A linear function is, as you say, one that satisfies $f(x+y) = f(x) + f(y)$ and $f(kx) = kf(x)$.
From this, one can show that (for real-valued functions of a real variable), the true linear functions are of the form $f(x) = ax$.  That is, linear functions are a special kind of affine function (the ones with $b = 0$).
Is all of this clear?
