On certain subclasses of Bazilevič functions (Q1409815)

For \(\alpha>0\) and \(\beta \in \mathbb{R}\) the class \(B(\alpha, \beta)\) of Bazilevich functions of type \((\alpha,\beta)\) is defined by the formula \[ f(z)=\left((\alpha +i\beta)\int_0^zp(\xi)(g(\xi))^{\alpha}\xi^{i\beta-1}d\xi\right)^{\frac{1}{\alpha+i\beta}}, \] where \(z \in \mathbb{D}=\left\{z: | z| <1\right\}\) and \(p(z)=1+p_1z+...\) is holomorphic in \(\mathbb{D}\) with \(\text{Re}\,p(z)>0,\) \(z \in \mathbb{D}\) and \(g(z)=z+b_2z^2+...\) is univalent and starlike in \(\mathbb{D}.\) The class \(B(\alpha, \beta)\) consists of univalent functions in \(\mathbb{D}.\) Taking instead of starlike functions \(g\) the different subclasses defined by the subordination \[ \frac{zg'(z)}{g(z)}\prec\Phi(z), \] \(z \in \mathbb{D}\) where \(\Phi\) is holomorphic in \(\mathbb{D}\), \(\Phi(0)=1\) and \(\Phi(\mathbb{D})\) is convex, the author obtains a corresponding class \(B_{\Phi}(\alpha, \beta).\) The paper contains several results about subordination and inclusion for the class \(B_{\Phi}(\alpha, \beta)\) under some extra assumptions for \(\Phi.\) As methods some results of St. Ruscheweyh and S. S. Miller and P. T. Mocanu are used.