  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)$.
