Question

Consider the following condition on a function \( f : {C} \to {C} \): \[ |f(z)| = 1 \quad {for all } z \in {C} { such that } \operatorname{Im}(z) = 0. \] Which one of the following is correct?

• A. There is a non-constant analytic polynomial \( f \) satisfying the condition.
• B. Every entire function \( f \) satisfying the condition is a constant function.
• C. Every entire function \( f \) satisfying the condition has no zeroes in \( {C} \).
• D. There is an entire function \( f \) satisfying the condition with infinitely many zeroes in \( {C} \).

Hint:

For modulus constraints, check the properties of entire functions and their behavior across the complex plane.

Answer 1

The Correct Option is C

Step 1: Analyzing the condition. The condition \( |f(z)| = 1 \) for all \( z \) with \( \operatorname{Im}(z) = 0 \) implies that \( f(z) \) has modulus 1 along the real axis. Step 2: Consequence of the condition. An entire function with modulus 1 on a line (e.g., the real axis) cannot have zeroes anywhere in \( {C} \), as this would contradict the modulus condition. Step 3: Conclusion. The correct statement is \( {(3)} \): Every entire function \( f \) satisfying the condition has no zeroes in \( {C} \).