Study analytical function subordination properties by applying a novel linear operator

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 $M_{\sigma ,\mu }^{n,\varsigma }{\rm Y}\left(s\right)$ . 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 $M_{\sigma ,\mu }^{n,\varsigma }{\rm Y}\left(s\right)$ .

A new operator $M_{\sigma ,\mu }^{n,\varsigma }{\rm Y}\left(s\right):\Lambda ⟶\Lambda$ has been established in this paper connected to the Dziok-Srivastava operator $T_{\sigma }^n$ and the Hadamard product corresponding to the Komatu integral operator ${\Omega }_{\mu }^{\varsigma }$ . The difference operator $M_{\sigma ,\mu }^{n,\varsigma }\Upsilon \left(s\right)$ can have specific properties derived by applying the differential subordination technique. And the objective of this paper is to make use of the connection $({\beta }_1\mu +1)M_{\sigma ,\mu }^{n+1,\varsigma }\Upsilon \left(s\right)=w{\left(M_{\sigma ,\mu }^{n,\varsigma }\Upsilon \left(s\right)\right)}^{\text{'}}+{\beta }_1\mu \left(M_{\sigma ,\mu }^{n,\varsigma }\Upsilon \left(s\right)\right).$