On certain Bazilević functions of order \(\beta\)
Let \(A(n)\) be the class of functions of the form \(f(z)=z+\sum_{k=n+1}^ \infty a_ k z^ k\), \(n\in\mathbb{N}\), which are analytic in the unit disk \(U\). A function \(f(z)\) in \(A(n)\) is said to be in the class \(B(n,\alpha,\beta)\) if it satisfies \[ \text{Re}\left\{{{f'(z)f(z)^{\alpha-1}} \over {z^{\alpha- 1}}}\right\}>\beta \qquad (z\in U) \] for some \(\alpha\) (\(\alpha>0\)) and \(\beta\) (\(0\leq\beta<1\)). The class \(B(n,\alpha,\beta)\) is the subclass of Bazilević functions in \(U\). Also \(f(z)\) in \(B(n,\alpha,\beta)\) is called to be a Bazilević function of order \(\beta\) in \(U\). The author derives some properties of functions belonging to the class \(B(n,\alpha,\beta)\).
scientific article; zbMATH DE number 89255