Some classes of analytic functions associated with conic regions
Let \(S\) denote the class of all functions \(f\) analytic and univalent in the open unit disk \(U\), normalized by \(f(0) = f'(0)-1 = 0\). For a fixed \(k \geq 0\) denote by \(k\)-UCV the class of all \(k\)-uniformly convex functions introduced and investigated by \textit{S. Kanas} and the reviewer [J. Comput. Appl. Math. 105, No. 1--2, 327--336 (1999; Zbl 0944.30008)]. Recall that a function \(f \in S\) is \(k\)-uniformly convex in \(U\) if it maps the intersection of \(U\) with any disk centered at the point \(\zeta\), where \(|\zeta| \leq k\), onto a convex domain. The class of \(k\)-uniformly convex functions can be defined equivalently as follows: a function \(f \in S\) belongs to the class \(k\)-UCV if and only if for all \(z\in U\), \[ \mathrm{Re}\left(1 + \frac{zf''(z)}{f'(z)}\right) > k\left|\frac{zf''(z)}{f'(z)}\right|. \] Let \(k\)-ST denote the class of functions associated with \(k\)-UST via the Alexander relation. The authors introduce some classes of functions which generalize the classes \(k\)-UCV and \(k\)-ST and are also related to conic domains. Let \(\alpha\), \(\beta\) and \(k\) be nonnegative real numbers such that \(0 \leq \beta < \alpha \leq 1\) and \(k(1-\alpha) < 1-\beta\). A function \(f \in S\) is said to be in the class \(k\text{-UCV}(\alpha,\beta)\) if it satisfies, for all \(z\in U\), the condition \[ \mathrm{Re}\left(1 + \frac{zf''(z)}{f'(z)}\right) - \beta > k\left|\frac{zf''(z)}{f'(z)} - \alpha\right|. \] The class \(k\text{-ST}(\alpha,\beta)\) is defined by the relation \(f \in k\text{-UST}(\alpha,\beta)\) if and only if \(zf'(z) \in k\text{-ST}(\alpha,\beta)\). In the paper under review the authors obtain many results in the considered classes in analogy to known results for the classes \(k\)-UCV and \(k\)-ST.
scientific article; zbMATH DE number 6019766