On normalized Rabotnov function associated with two subclasses of analytic functions with positive coefficients

Let \({\mathbb {R}}_{\alpha ,\beta }(z)=z+{\displaystyle \sum \limits _{n=2}^{\infty }}\frac{\beta ^{n-1}\Gamma (1+\alpha )}{\Gamma ((1+\alpha )n)}z^{n}\) ${\mathbb{R}}_{\alpha ,\beta }(z)=z+{\displaystyle \sum _{n=2}^{\infty }}\frac{{\beta }^{n-1}\mathrm{\Gamma }(1+\alpha )}{\mathrm{\Gamma }((1+\alpha )n)}{z}^n$ be the normalized Rabotnov functions. The purpose of the present paper is to determine necessary and sufficient conditions and inclusion relation for the normalized Rabotnov function \({\mathbb {R}}_{\alpha ,\beta }(z)\) ${\mathbb{R}}_{\alpha ,\beta }(z)$ to be in two subclasses of analytic functions with positive coefficients. Further, we consider an integral operator related to the Rabotnov function \({\mathbb {R}}_{\alpha ,\beta }(z)\) ${\mathbb{R}}_{\alpha ,\beta }(z)$ . Several examples of the main results are also considered.