Constant Complex-Valued Functions
Posted: Sun Jul 10, 2016 3:56 pm
(1) Suppose that \( \displaystyle f : A \subset \mathbb{C} \longrightarrow \mathbb{C} \) is an analytic function on the open connected set \( \displaystyle A \) and that \( \displaystyle f(z) \) is real for all \( \displaystyle z \in A \). Show that \( \displaystyle f \) is constant.
(2) Suppose that \( \displaystyle f : A \subset \mathbb{C} \longrightarrow \mathbb{C} \) is an analytic function on the open connected set \( \displaystyle A \) and that \( \displaystyle f(A) \subset \left\{ z \in \mathbb{C} \big| |z|=3 \right\} \). Show that \( \displaystyle f \) is constant.
(3) Suppose that \( \displaystyle f : \mathbb{C} \longrightarrow \mathbb{C} \) is continuous and that \( \displaystyle f(z) = f(2z) \, , \, \forall z \in \mathbb{C} \). Show that \( \displaystyle f \) is constant.
(2) Suppose that \( \displaystyle f : A \subset \mathbb{C} \longrightarrow \mathbb{C} \) is an analytic function on the open connected set \( \displaystyle A \) and that \( \displaystyle f(A) \subset \left\{ z \in \mathbb{C} \big| |z|=3 \right\} \). Show that \( \displaystyle f \) is constant.
(3) Suppose that \( \displaystyle f : \mathbb{C} \longrightarrow \mathbb{C} \) is continuous and that \( \displaystyle f(z) = f(2z) \, , \, \forall z \in \mathbb{C} \). Show that \( \displaystyle f \) is constant.