Third-Order Differential Subordination and Superordination Results for Meromorphically Multivalent Functions Associated with the Liu-Srivastava Operator. (8th July 2014) - Tang, Huo; Srivastava, H. M.; Li, Shu-Hai; Ma, Li-Na
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 planeC . Also letp be analytic in the unit diskU = z : z ∈ C and z < 1 and suppose thatψ : C 4 × U → C . In this paper, we investigate the problem of determining properties of functionsp ( z ) that satisfy the following third-order differential superordination:Ω ⊂ ψ p z, z p ′ z, z 2 p ′ ′ z, z 3 p ′ ′ ′ z ; z : z ∈ 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.