List of top Complex Analysis Questions on Mobius transformations

Let \(f(z) = \dfrac{z}{1-z}\) and \(g(z) = \dfrac{1+z}{1-z}\) be two Mobius transformations defined on the unit disc \(D = \{z \in \mathbb{C} : |z| < 1\}\). Consider the following statements:
\(S_1\): \(f(D) \subseteq g(D)\)
\(S_2\): \(g(D) \subseteq f(D)\)
Which of the following statements is/are CORRECT?

Let \( T \) be a Möbius transformation such that \( T(0) = \alpha, T(\alpha) = 0 \) and \( T(\infty) = -\alpha \), where \( \alpha = \frac{-1 + i}{\sqrt{2}} \). Let \( L \) denote the straight line passing through the origin with slope \( -1 \), and let \( C \) denote the circle of unit radius centered at the origin. Then, which of the following statements are TRUE?

Let \( T(z) = \frac{az + b}{cz + d}, ad - bc \neq 0 \), be the Möbius transformation which maps the points \( z_1 = 0, z_2 = -i, z_3 = \infty \) in the z-plane onto the points \( w_1 = 10, w_2 = 5 - 5i, w_3 = 5 + 5i \) in the w-plane, respectively. Then the image of the set \( S = \{ z \in \mathbb{C} : \text{Re}(z) < 0 \ \) under the map \( w = T(z) \) is}