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.
Best Answer
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).