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

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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