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Third-order differential subordination and superordination results for meromorphically multivalent functions associated with the Liu-Srivastava operator. (English) Zbl 1474.30128

Summary: There are many articles in the literature dealing with the first-order and the second-order differential subordination and superordination problems for analytic functions in the unit disk, but only a few articles are dealing with the above problems in the third-order case (see, e.g., [J. A. Antonino and S. S. Miller, Complex Var. Elliptic Equ. 56, No. 5, 439–454 (2011; Zbl 1220.30035)] and [S. Ponnusamy and O. P. Juneja, in: Current topics in analytic function theory. Singapore: World Scientific. 274–290 (1992; Zbl 0991.30012)]). The concept of the third-order differential subordination in the unit disk was introduced by Antonino and Miller [loc. cit.]. Let \(\Omega\) be a set in the complex plane \(\mathbb{C}\). Also let \(\mathfrak{p}\) be analytic in the unit disk \(\mathbb{U} = \left\{z : z \in \mathbb{C}\text{ and }\left|z\right| < 1\right\}\) and suppose that \(\psi : \mathbb{C}^4 \times \mathbb{U} \rightarrow \mathbb{C}\). In this paper, we investigate the problem of determining properties of functions \(\mathfrak{p}(z)\) that satisfy the following third-order differential superordination: \(\Omega \subset \left\{\psi \left(\mathfrak{p} \left(z\right), z \mathfrak{p}' \left(z\right), z^2 \mathfrak{p}'' \left(z\right), z^3 \mathfrak{p}''' \left(z\right); z\right) : z \in \mathbb{U}\right\}\). As applications, we derive some third-order differential subordination and superordination results for meromorphically multivalent functions, which are defined by a family of convolution operators involving the Liu-Srivastava operator. The results are obtained by considering suitable classes of admissible functions.

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination
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