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 planeC. Also letpbe analytic in the unit diskU=z:z∈C and z<1and suppose thatψ:C4×U→C. In this paper, we investigate the problem of determining properties of functionsp(z)that satisfy the following third-order differential superordination:Ω⊂ψpz,zp′z,z2p′′z,z3p′′′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.

Third-order differential sandwich-type results involving the Liu-Owa integral operator

“…Let Q be the set of analytic and univalent functions q on the set U \ E(q), where E(q) = {ξ : ξ ∈ ∂U and lim z→ξ q(z) = ∞}, and are such that min |q (ξ)| = ρ > 0 for ξ ∈ ∂U \ E(q). Further, let the subclass of Q for which q(0) = a be written by Q(a) and 6,32,34]). Let ψ : C 4 × U → C and h(z) be univalent in U.…”

“…On the other word, Tang et al [33] (see also [34]) obtained the following result for the class of admissible functions Ψ n [Ω, q].…”

“…Later on, Tang et al [33] introduced the notion of third-order differential superordination and also studied the corresponding third-order differential superordination involving the generalized Bessel functions. Based on [6] and [33], Tang et al [34] considered third-order differential subordination and superordination results for meromorphically multivalent functions associated with the Liu-Srivastava operator. Here, in the present paper, we aim to study third-order differential sandwichtype results of multivalent analytic functions involving the Liu-Owa integral operator Q α β,p defined by (1.2).…”

Third-order differential subordination and superordination involving a fractional operator

“…They determined the properties of p functions that satisfy the following third-order differential subordination: f .p.z/; zp 0 .z/; z 2 p 00 .z/; z 3 p 000 .z/I z/ W z 2 Ug : In 2013, Jeyaraman et al [33] also applied the third-order subordination result on the Schwarzian derivative. In 2014, Tang et al [34] introduced the concept of the third-order differential superordination, which is a generalization of the second-order differential superordination. They determined the properties of functions that satisfy the following third-order differential superordination:…”

“…In 2014, Farzana et al [37] discussed some third-order differential subordination results for analytic functions which are associated with the fractional derivative operator. The present study utilized the methods of the third-order differential subordination and superordination results of Antonino and Miller [26] and Tang et al [34], respectively. Certain suitable classes of admissible functions are considered in this study, and some applications of the third-order differential subordination and superordination of analytic functions associated with the new operator are investigated.…”