In this paper, we use the concept of $q$-calculus in geometric function theory. For some $\alpha$, $\alpha\in [0,1)$, we consider normalized analytic functions $f$ such that $f’(z)/{\rm d}_qf(z)$ lies in half-plane $\{w:\mathfrak {Re}\ w>\alpha\}$ for all $z$, $|z|< 1$. Here ${\rm d}_q$ is the Jackson $q$-derivative operator well-known in the $q$-calculus theory. The paper is devoted to the coefficient problems of such functions for real and for complex numbers $q$. Coefficient bounds are of particular interest, because of them some geometrical properties of the function can be obtained.