Tests for certain rational functions to belong to the class \(K_ n(D)\) (Q795200)

Let D be a domain in \({\mathbb{C}}\) and n a natural number. Then a single valued function F(z) is said to belong to the class \(K_ n(D)\) if it is holomorphic in D and the n-th divided difference \([F(z);z_ 0z_ 1...z_ n]\) is not equal to zero for any \(z_ 0,z_ 1,...,z_ n\in D.\) It is noticed that \(K_ 1(D)\) is the set of all holomorphic schlicht functions. In this paper, the authors give various kinds of necessary and sufficient conditions for some simple rational functions to belong to the class \(K_ n(D)\) where D is the unit disk E or a sector \(U_ p=\{z\in {\mathbb{C}}| | \arg z|<\pi /(p+1)\}.\)