Inclusion properties for classes of analytic functions associated with conic domains
Let \(h_k\) (\(0\leq k< \infty\)) be a function extremal in a subclass of the class of Carathéodory functions, related to conic domains. The notion of convexity and starlikeness related to conic domains was introduced by \textit{S. Kanas} and \textit{A. Wiśniowska} [J. Comput. Appl. Math. 105, No. 1--2, 327--336, (1999; Zbl 0944.30008); Rev. Roum. Math. Pures Appl. 45, No. 4, 647--657, (2000; Zbl 0990.30010)] and developed later by \textit{S. Kanas} and \textit{T. Sugawa} [Ann. Acad. Sci. Fenn. Math. 31, No. 2, 329--348, (2006; Zbl 1098.30011)], \textit{H. M. Srivastava} [Gen. Math. 15, No. 2--3, 201--226, (2007; Zbl 1199.30107)], \textit{K. I. Noor} et al. [Complex Var. Elliptic Equ. 61, No. 10, 1418--1433, (2016; Zbl 1360.31002)] and others.NEWLINENEWLINENEWLINEThe author denotes by \(\mathcal {P}_\gamma (a,k)\;(\gamma \geq 2)\) the class of functions \(q(z) = 1+c_1z+\cdots \) such that NEWLINE\[NEWLINE q(z) = a+\frac {1-a}{2}\int \limits_0^{2\pi}h_k(re^{-it})\,{\operatorname {d}}m(t), NEWLINE\]NEWLINE where \(m\) satisfies NEWLINE\[NEWLINE \int \limits_0^{2\pi}{\operatorname {d}}m(t) = 2\quad\text{and}\quad \int \limits_0^{2\pi}| \,{\operatorname {d}}m(t)| \leq \gamma. NEWLINE\]NEWLINENEWLINENEWLINEAlso, let \(\mathcal {B}_\gamma (a,k)\) be defined by NEWLINE\[NEWLINE \mathcal {B}_\gamma (a,k)=\left \{\left (\frac {\gamma}4+\frac 12\right)q_1+\left (\frac {\gamma}4-\frac 12\right)q_2:\;q_1, q_2\prec h_{a,k}\right \}, NEWLINE\]NEWLINE where \(h_{a,k} = a+(1-a)h_k\). The author gives a characterization of the classes \(\mathcal {P}_\gamma (a,k)\) and \(\mathcal {B}_\gamma (a,k)\) with inclusion properties of some related classes of analytic functions.
scientific article; zbMATH DE number 6589830