Some classes of analytic functions associated with conic regions (Q763744)

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.