We present several new results for fourth-order differential subordination and superordination in this paper by using the differential linear operator . Relevant connections between the new results presented here and those considered in previous works are addressed. The properties and results concerning the differential subordination theory are symmetric to the properties obtained using the differential superordination theory, and by combining them, sandwich-type theorems are obtained.

The objective of this research paper is to show how the Bennan'sconjecture  become a useful tool  to construct a holomorphic function on the cardioid domain, and on the boundary of unit disk. Moreover , we have addressed some applications on the existence of cusp on the boundary of arising from integrability of conformalmaps through one of the polar function in the general solution of Laplace equation.

The purpose of this research paper, is to present the second- order homogeneous complex differential equation f" + H(z)f = 0, which defined on the disk D = {z ∈ ℂ : |z − i| ≤ 1} ⊆ ℂ, where H(z) = e p(z), to show it an invariant by applying Liouville and self-adjoint transformation with an examine the convexity property of its coefficient H(z) = e p(z), in order to study the growing and bounded solution of consider equation.

In this research paper, we explain the use of  the convexity and the starlikness properties of  a given function  to generate  special properties of differential subordination and superordination functions in the classes of analytic functions that have  the form   in the unit disk. We also show the significant of  these properties to derive sandwich results when the Srivastava- Attiya operator   is used.

No abstract