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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 Omega be a set in the complex plane C. Also let p be analytic in the unit disk U = {z : z is an element of C and |z| < 1} and suppose that psi : C-4 x U -> C. In this paper, we investigate the problem of determining properties of functions p(z) that satisfy the following third-order differential superordination: Omega subset of {psi(p (z), zp' (z), z(2)p " (z), z(3)p"' (z) ; z) : z is an element of 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.

abstract and applied analysis

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