Some relations between certain classes consisting of \(\alpha \)-convex type and Bazilević type functions
Using the more general method of differential subordinations founded by \textit{S. S. Miller} and \textit{P. T. Mocanu} [Differential subordinations: theory and applications. New York, NY: Marcel Dekker (2000; Zbl 0954.34003)], several inclusion relations between certain classes consisting of \(\alpha\)-convex type functions are obtained, and various interesting consequences are derived. In particulary, the following theorem is proved: Let \(f(z)= z+ a_{n+1} z^{n+1}+ a_{n+2} z^{n+2}+\cdots\) with \(f'(z)\cdot f(z)/z\neq 0\) for all \(z\in{\mathcal U}= \{z:|z|< 1\}\), and let \(\delta\in\mathbb{R}\), \(\mu\in\mathbb{R}\). If \[ \text{Re}\Biggl\{\mu{zf'(z)\over f(z)}+ \delta\Biggl(1+ {zf''(z)\over f'(z)}\Biggr)\Biggr\}<\delta+ \mu+ {n\cdot M\over M+1},\quad z\in{\mathcal U}, \] where \(M> 1\), then \[ \Biggl| [f'(z)]^\delta\Biggr[{f(z)\over z}\Biggr]^\mu- 1\Biggr|< M,\quad z\in{\mathcal U}, \] where the powers are the principal ones.
scientific article; zbMATH DE number 5992301