Third-Order Differential Subordination and Superordination Results for Meromorphically Multivalent Functions Associated with the Liu-Srivastava Operator
Date
2014
Authors
Tang, Huo
Srivastava, Hari M.
Li, Shu-Hai
Ma, Lin-Na
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Publisher
Abstract and Applied Analysis
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 ℂ. Also let P be analytic in the unit disk U = { Z ∶ Z ∈ ℂ and |Z| < 1} and suppose that ψ ∶ ℂ^4 × U → ℂ. In this paper, we investigate the problem of determining properties of functions P(Z) that satisfy the following third-order differential superordination: Ω ⊂ {ψ(P(Z), ZP' (Z), Z^2P''(Z), Z^3P'''(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.
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Citation
Tang, H., Srivastava, H.M., Li, S., & Ma, L. (2014). Third-Order Differential Subordination and Superordination Results for Meromorphically Multivalent Functions Associated with the Liu-Srivastava Operator. Abstract and Applied Analysis, Vol. 2014, Article ID 792175.