On subclasses of close-to-star functions of order $\mu$ and type $(\alpha,\beta)$

Year 2025, Volume: 74 Issue: 3, 546 - 559, 23.09.2025

Ramalingam Sathish Srinivasan , Raman Ezhilarasi , Ayotunde Olajide Lasode , Thirumalai Vinjimur Sudharsan

https://doi.org/10.31801/cfsuasmas.1541978

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.

Keywords

Close-to-star function , starlike function , Carlson-Shaffer operator , integral representation , coefficient estimate , Fekete-Szegö estimate , inclusion property

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