We present several new results for higher-order (fourth-order) differential subordination and superordination in this paper by using the new operator Hα,β,𝛿,𝜉,γ,nf(v) and offer numerous new findings for fourth-order differential subordination and superordination The innovative discoveries presented here are connected to those mentioned in previous articles. The differential subordination theory’s characteristics and outcomes are symmetric to the properties gained. Sandwich-type theorems are created by merging differential superordination theory with sandwich-type theorems.

Th  goal of the pr s nt p p r is to obt in some differ tial sub rdin tion an  sup r dination the rems for univalent functions related b  differential operator  Also, we discussed some sandwich-type results.

In this paper, we defined and studied some types of ideals of B-algebra, which we called them, Complete Ideal, Closed Complete Ideal, n-Ideal and Closed n-Ideal respectively. In addition, we gave some propositions that explained some relationships between these ideals types.

In this paper, we have found the solution of second-order delay differential equations of retarded type with multiple delays. As well as developing an approximation for finding characteristic roots for such delay differential equations via the method of spectral tau which depends on the basis mixed Fourier basis or shifted Chebyshev polynomials.

This paper aims to find new analytical closed-forms to the  solutions of the nonhomogeneous functional differential equations of the nth order with finite and constants delays and various initial delay conditions in terms of elementary functions using Laplace transform method. As well as, the definition of dynamical systems for ordinary differential equations is used to introduce the definition of dynamical systems for delay differential equations which contain multiple delays with a discussion of their dynamical properties: The exponential stability and strong stability