A Novel Class of Starlike Functions

Abstract

In the past several subclasses of starlike functions are defined involving real part and modulus of certain expressions of functions under study, combined by way of an inequality. In the similar fashion, we introduce a new class $\mathcal{S}^*_{q}(\alpha)$, consisting of normalized analytic univalent functions $f$ in the open unit disk $\mathbb{D}$, satisfying $$\RE\left(\dfrac{z f'(z)}{f(z)}\right) \geq \left|1+\dfrac{z f''(z)}{f'(z)} -\dfrac{z f'(z)}{f(z)}-\alpha\right| \quad (0 \leq \alpha <1).$$ Evidently, $\mathcal{S}^*_{q}(\alpha) \subset \mathcal{S}^*$, the class of starlike functions. We first establish $\mathcal{S}^*_{q}(\alpha) \subset \mathcal{S}^*(q_\alpha)$, the class of analytic functions $f$ satisfying $z f'(z)/f(z)\prec q_\alpha(z),$ where $q_\alpha$ is an extremal function. Some necessary and sufficient conditions for functions belonging to these classes are obtained in addition to the inclusion and radius problems. Further, we estimate logarithmic coefficients, inverse coefficients and Fekete-Szeg\"o functional bounds for functions in $ \mathcal{S}^*(q_\alpha)$.