The intention of the current investigation is to demonstrate that if an additive mapping $\mathcal{H}:R\to R$ fulfills any one of the following identities:\begin{enumerate}\item [$(i)$] $3\mathcal{H}(r^{3p})=\mathcal{H}(r^p)\varphi(r^{2p})+\varphi(r^p) \mathcal{H}(r^{p})\varphi(r^p)+ \varphi(r^{2p})\mathcal{H}(r^p)$\item [$(ii)$] $2\mathcal{H}(r^{2p})=\mathcal{H}(r^p)\varphi(r^{p})+\varphi(r^p)\mathcal{H}(r^{p})$\item [$(ii)$] $\mathcal{H}(r^{3p})=\varphi(r^p) \mathcal{H}(r^{p}) \varphi(r^p)$ for all $r\in R$,\end{enumerate}then $\mathcal{H}$ is a $\varphi$-centralizer on $R$, where $R$ is any suitable, torsion-free semiprime ring and $p$ is a fixed integer greater than or equal to 1. As a result of the primary theorems, involution $I_v$ related observations are also provided. We will also consider criticism and discussion alongside the proofs of theorems. Suitable examples given in favor of justification.

The aim of this paper is to investigate various subfamilies of analytic functions to find inclusion properties, and necessary and sufficient conditions for the Rabotnov function to be in these subfamilies. Furthermore, several corollaries will be implied from our main results.

In this paper, the subfamilies Cκ3 (κ1,κ2) and S∗ κ3 (κ1,κ2) investigated through the Hohlov operator. More specifically, a number of sufficient requirements are given in order for the aforementioned functions subfamilies. Moreover, our results will imply several corollaries.