@article{article_1541978, title={On subclasses of close-to-star functions of order $\mu$ and type $(\alpha,\beta)$}, journal={Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics}, volume={74}, pages={546–559}, year={2025}, DOI={10.31801/cfsuasmas.1541978}, author={Srinivasan, Ramalingam Sathish and Ezhilarasi, Raman and Lasode, Ayotunde Olajide and Sudharsan, Thirumalai Vinjimur}, keywords={Close-to-star function, starlike function, Carlson-Shaffer operator, integral representation, coefficient estimate, Fekete-Szegö estimate, inclusion property}, abstract={Let $\mathcal{P}_{\mu}$ represent the class of analytic functions $\wp(z)$ defined in the open unit disc $\varDelta=\{z: |z|<1 \}$ with $\wp(0)=1$ and $$ \left| \frac{\wp(z)-1}{\wp(z)+1} \right| < \mu. $$ In this paper, we introduce two new subclasses $\mathcal{L}_{u,v}(\alpha,\beta,\mu)$ and $\mathcal{L}^\lambda_{u,v}(\alpha,\beta,\mu)$ of the class of close-to-star functions that satisfy the conditions: $$ \left( \alpha \frac{(\mathscr{L}_{u,v} f(z))’}{g’(z)}+\beta \frac{\mathscr{L}_{u,v} f(z)}{g(z)} \right) \in\mathcal{P}_{\mu} $$ and $$ \left(\alpha \frac{((\mathscr{L}_{u,v} f(z))’)^{\lambda }{(g’(z))^{\lambda }+\beta \frac{(\mathscr{L}_{u,v} f(z))^{\lambda }{(g(z))^{\lambda } \right) \in\mathcal{P}_{\mu}, $$ respectively. Functions $f$ in the new classes are normalized analytic functions defined in the unit disc $\varDelta$ such that $g$ is starlike and $\mathscr{L}_{u,v}$ is the Carlson-Shaffer operator. Some reported results for $f\in\mathcal{L}_{u,v}(\alpha,\beta,\mu)$ include the integral representation formula, some coefficient estimates, Fekete-Szegö estimates for real and complex parameters, and some inclusion properties. All the results are sharp. Again, some early coefficient estimates for functions $f\in\mathcal{L}^\lambda_{u,v}(\alpha,\beta,\mu)$ are investigated. Furthermore, a number of remarks to show the relationship between the new classes and some existing classes are clearly discussed.}, number={3}, publisher={Ankara University}