What is the use of introduction of term univalent function? > What is the use of defining a new term univalent function although there already existed a definition of bijective, one-to-one functions ?

Well let's at least look at the definition. The definition has 3 parts (in bold) to it, so it is more convenient to create a definition instead of writing it out every time:

> A _univalent_ function is a **holomorphic** function on an **open subset** of $\mathbb{C}$ that is **injective** (one-to-one)

So they are functions who, on some open subset $U$ of the complex plane, satisfy these strong conditions:

1) Holomorphic (Differentiable in a neighborhood around every point in $U$)

2) Injective (One-to-one)