Univalent Functions

An analytic one-to-one function \(f:\mathbb {D}\rightarrow \mathbb {C}\) is called a univalent function. The nth Taylor coefficient \(a_n=f^{(n)}(0)/n!\) of a univalent function f satisfies the sharp inequality \(|a_n|\leqslant n\) . Bieberbach conjectured this inequality and proved it for \(n=2\) . In this chapter, the coefficient result \(|a_2|\leqslant 2\) is proved which leads to several significant consequences. Several other basic properties of univalent functions are also explored.