A $q$-generalization of class $R$

For $0 < q < 1$, we define a new class $R_q$ of analytic functions using $q$-difference operator, which is a $q$-analogue of class $R$. We prove that the class $R$ is properly contained in $R_q$ and find a sharp lower bound for the real part of $D_q f$, where $f \in R_q$. We investigate certain convolution properties of functions in the class $R_q$ including that the class $R_q$ is closed under convolution for a definite range of $q$. The findings of the present manuscript essentially generalizes some well-known results in the literature.