Teaching Haryana (HPSC) Assistant Professor Mock Test 2025 Mathematical Science Complex Analysis Theorems of Complex analysis
Consider a function f that is holomorphic and non-constant on a domain \(D = \{z \in \mathbb{C} : |z| < 2\}\) and continuous on its closure \( \overline{D}\) . Suppose that |f(z)| attains its maximum value on the boundary of D, except at z = i , where |f(i)| = |f(1+i)| but |f(1+i)| is not the maximum on the boundary. Which of the following statement must be true according to the Maximum Modulus Principle?
1
|f(z)| must be constant throughout D because it achieves a local maximum inside D.
2
|f(z)| cannot attain a maximum value at any point inside unless f is a constant function.
3
f must have a singularity at z = i because |f(i)| equals |f(1+i)| without being the boundary maximum.
4
f is not holomorphic at z = i since |f(i)| = |f(1+i)| and does not adhere to the typical behavior expected by holomorphic functions at a boundary point.
5
Question Not Attempted