Let $f ( z ) = \sum _ { n \geq 0 } a _ { n } z ^ { n }$ be an analytic function on the open unit disc $D$ around 0 with $a _ { 1 } \neq 0$. Suppose that $\sum _ { n \geq 2 } \left| n a _ { n } \right| < \left| a _ { 1 } \right|$. Then which of the following are true? (A) There are only finitely many such $f$. (B) $\left| f ^ { \prime } ( z ) \right| > 0$ for all $z \in D$. (C) If $z , w \in D$ are such that $z \neq w$ and $f ( z ) = f ( w )$, then $a _ { 1 } = - \sum _ { n \geq 2 } a _ { n } \left( z ^ { n - 1 } + z ^ { n - 2 } w + \cdots + w ^ { n - 1 } \right)$. (D) $f$ is one-one on $D$.
Let $f ( z ) = \sum _ { n \geq 0 } a _ { n } z ^ { n }$ be an analytic function on the open unit disc $D$ around 0 with $a _ { 1 } \neq 0$. Suppose that $\sum _ { n \geq 2 } \left| n a _ { n } \right| < \left| a _ { 1 } \right|$. Then which of the following are true?\\
(A) There are only finitely many such $f$.\\
(B) $\left| f ^ { \prime } ( z ) \right| > 0$ for all $z \in D$.\\
(C) If $z , w \in D$ are such that $z \neq w$ and $f ( z ) = f ( w )$, then $a _ { 1 } = - \sum _ { n \geq 2 } a _ { n } \left( z ^ { n - 1 } + z ^ { n - 2 } w + \cdots + w ^ { n - 1 } \right)$.\\
(D) $f$ is one-one on $D$.