I want to continue the topic “Schwarz Reflection Principle and several applications”. Today we discuss the following.
Question. If entire function satisfies whenever , then there exist a non-negative integer number and a constant satisfying such that .
Solution. Suppose has a zero at the origin of order for some and the other zeros of in the unit disk are listed as , repeating as necessary to account for multiplicities. The set of zeros in the unit disk must be finite since that set cannot have any limit points.
Note that for all . Let . Then is an entire function with no zeros in the unit disk and with the property that for .
By the maximum modulus principle and must both achieve their maximum values for the unit disk on the boundary, from which we conclude that for all . But then, that means that must be constant, with .
So we have that
But that expression has poles whenever . The only way we could have be an entire function is for there to be no such . Hence we conclude that for some nonnegative .
Here is an even further simplification.
Dividing by some finite power of (which does not change the property), we can assume that . Now consider
They are meromorphic on and coincide on the unit circle. Thus they coincide everywhere. Hence
so is bounded and, thereby, constant.
I will provide another proof mainly based on the Schwarz Reflection Principle. However, to this purpose, an extension of Morera’s Theorem for toy contours should be introduced firstly.