Abstract
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.