## Abstract and Applied Analysis

### Third-Order Differential Subordination and Superordination Results for Meromorphically Multivalent Functions Associated with the Liu-Srivastava Operator

#### Abstract

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., Antonino and Miller (2011) and Ponnusamy et al. (1992)). The concept of the third-order differential subordination in the unit disk was introduced by Antonino and Miller in (2011). Let Ω be a set in the complex plane $\mathbb{C}$. Also let $\mathfrak{p}$ be analytic in the unit disk $\mathbb{U}=\{z:z\in \mathbb{C}\text{\hspace\{0.17em\}\hspace\{0.17em\}and\hspace\{0.17em\}\hspace\{0.17em\}}|z|<1\}$ and suppose that $\psi :{\mathbb{C}}^{4}{\times}\mathbb{U}\to \mathbb{C}$. In this paper, we investigate the problem of determining properties of functions $\mathfrak{p}(z)$ that satisfy the following third-order differential superordination: $\mathrm{\Omega }\subset \{\psi (\mathfrak{p}(z),z{\mathfrak{p}}^{\prime }(z),{z}^{2}{\mathfrak{p}}^{\prime \prime }(z),{z}^{3}{\mathfrak{p}}^{\prime \prime \prime }(z);z):z\in \mathbb{U}\}$. 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.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 792175, 11 pages.

Dates
First available in Project Euclid: 6 October 2014

https://projecteuclid.org/euclid.aaa/1412605986

Digital Object Identifier
doi:10.1155/2014/792175

Mathematical Reviews number (MathSciNet)
MR3232865

Zentralblatt MATH identifier
07023078

#### Citation

Tang, Huo; Srivastava, H. M.; Li, Shu-Hai; Ma, Li-Na. Third-Order Differential Subordination and Superordination Results for Meromorphically Multivalent Functions Associated with the Liu-Srivastava Operator. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 792175, 11 pages. doi:10.1155/2014/792175. https://projecteuclid.org/euclid.aaa/1412605986

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