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 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: $\mathrm{\Omega }\subset \left\{\psi \left(\mathfrak{p}\left(z\right),z{\mathfrak{p}}^{\prime }\left(z\right),z^2{\mathfrak{p}}^{\prime \prime }\left(z\right),z^3{\mathfrak{p}}^{\prime \prime \prime }\left(z\right);z\right):z\in \mathbb{U}\right\}$. 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.