On \(\alpha\)-convex functions of order \(\beta\)
If \(f(z) = z + a_2 z^2+\dots\) is analytic in the unit disc \(U\) and \(f(z)f'(z)/z \neq 0\) in \(U\), let \(0 \leq \alpha \leq 1\) and define \(\Phi_\alpha(z) = \alpha (1 + zf'(z)/f''(z)) + (1 - \alpha) z f'(z)/f(z)\). The class of starlike functions of order \(\beta\) (\(0 \leq \beta \leq 1\)) is defined by \(\text{Re }\Phi_0(z) > \beta\), and the class of convex functions of order \(\beta\) by \(\text{Re }\Phi_1(z)> \beta\) for all \(z \in U\). This paper discusses and proves some class inclusion properties of the classes of \(\alpha\)-convex functions of order \(\beta\), type 1 defined by \(\text{Re }\Phi_\alpha(z) > \beta\) and of type 2 defined by \(\text{Re }\Phi_\alpha(z) < \beta\) in \(U\). The type 1 class with \(\beta = 0\) was introduced in 1969 by Mocanu. It should be noted that \(f''(z)\) is misprinted as \(f'(z)\) throughout the paper. Also, \(P\) and \(p\) seem to be used interchangeably.
scientific article; zbMATH DE number 1127594
scientific article