Question

Let \( \Omega \) be a non-empty open connected subset of \( \mathbb{C} \) and \( f: \Omega \to \mathbb{C} \) be a non-constant function. Let the functions \( f^2: \Omega \to \mathbb{C} \) and \( f^3: \Omega \to \mathbb{C} \) be defined by \[ f^2(z) = (f(z))^2 \quad {and} \quad f^3(z) = (f(z))^3, \quad z \in \Omega. \] 
Consider the following two statements: 
S1: If \( f \) is continuous in \( \Omega \) and \( f^2 \) is analytic in \( \Omega \), then \( f \) is analytic in \( \Omega \). 
S2: If \( f^2 \) and \( f^3 \) are analytic in \( \Omega \), then \( f \) is analytic in \( \Omega \). Then, which one of the following is correct?

• A. S1 is TRUE and S2 is FALSE
• B. S2 is TRUE and S1 is FALSE
• C. both S1 and S2 are TRUE
• D. neither S1 nor S2 is TRUE

Hint:

For functions that are powers of analytic functions, if the powers are analytic, the original function must also be analytic. This is a fundamental result in complex analysis.

Answer 1

The Correct Option is C

To solve this problem, we need to evaluate the truth of two statements concerning the analyticity of functions related to a non-empty open connected subset \( \Omega \) of the complex plane and a non-constant function \( f: \Omega \to \mathbb{C} \).

1. Statement S1: If \( f \) is continuous in \( \Omega \) and \( f^2 \) is analytic in \( \Omega \), then \( f \) is analytic in \( \Omega \).

Explanation:

• Since \( f^2(z) = (f(z))^2 \), if \( f^2 \) is analytic in \( \Omega \), its derivative exists.
• Let's compute the derivative of \( f^2 \): \(\frac{d}{dz}[f^2(z)] = 2f(z)f'(z)\). For this expression to be analytic (and thus have a derivative everywhere in \( \Omega \)), both \( f(z) \) and \( f'(z) \) must be well-defined. Given that \( f(z) \) is continuous, this implies \( f'(z) \) exists.
• Therefore, \( f \) must be analytic in \( \Omega \) since its derivative \( f'(z) \) exists and is continuous.

Thus, Statement S1 is TRUE.

2. Statement S2: If \( f^2 \) and \( f^3 \) are analytic in \( \Omega \), then \( f \) is analytic in \( \Omega \).

Explanation:

• For \( f^3(z) = (f(z))^3 \) to be analytic, its derivative must exist: \(\frac{d}{dz}[f^3(z)] = 3(f(z))^2f'(z)\).
• If both \( f^2(z) \) and \( f^3(z) \) are analytic, then similar reasoning to Statement S1 applies, where \(\frac{d}{dz}[f^2(z)] = 2f(z)f'(z)\) and \(\frac{d}{dz}[f^3(z)] = 3(f(z))^2f'(z)\) exist.
• This situation enforces \( f'(z) \) to be consistently defined, indicating that \( f \) is indeed analytic in \( \Omega \).

Thus, Statement S2 is TRUE.

In conclusion, both statements are true based on the rules of analytic functions in complex analysis. Therefore, the correct answer choice is:

both S1 and S2 are TRUE