# Study analytical function subordination properties by applying a novel linear operator

## TL;DR

This paper introduces a new linear operator and uses differential subordination to study properties of analytic univalent functions in complex analysis.

## Contribution

A novel linear operator is introduced and analyzed for its subordination and superordination properties in complex function theory.

## Key findings

- The new operator is connected to the Dziok-Srivastava and Komatu integral operators.
- Differential subordination techniques reveal properties of superordination and subordination.
- The operator's properties are derived using the Hadamard product and univalent function classes.

## Abstract

The study of theory for analytic univalent and multivalent functions is an old subject in mathematics, particularly in complex analysis, that has captivated a great deal of scholars owing to the sheer sophistication of its geometrical features as well as its many research possibilities. The study of univalent functions is one of many significant elements of complex analysis for both single and multiple variables. Investigators have become keen on the conventional investigation of this topic since at least 1907. Numerous scholars in the area of complex analysis have emerged since then, including Euler, Gauss, Riemann, Cauchy, and other people. Geometric function theory combines geometry and analysis.

This study employs the differential subordination technique to derive multiple characteristics from the new linear operator

Mσ,μn,ςΥ(s)
. The concept of the differential subordination subclass of analytical univalent functions is analyzed.

In this section, We studied some results on differential subordination and superordination using a specific class of univalent functions stated on a specific space of univalent functions stated on the open unit disc. Using properties of the operator, we discovered a number of properties of superordinations and subordinations related to the idea of the Hadamard product. We investigated several aspects of superordinations and subordinations using a new operator

Mσ,μn,ςΥ(s)
.

A new operator

Mσ,μn,ςΥ(s):Λ⟶Λ
has been established in this paper connected to the Dziok-Srivastava operator

Tσn
and the Hadamard product corresponding to the Komatu integral operator

Ωμς
. The difference operator

Mσ,μn,ςϒ(s)
can have specific properties derived by applying the differential subordination technique. And the objective of this paper is to make use of the connection

(β1μ+1)Mσ,μn+1,ςϒ(s)=w(Mσ,μn,ςϒ(s))′+β1μ(Mσ,μn,ςϒ(s)).