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For every 0 < q < 1 and 0 ≤ α < 1, we introduce a class of analytic functions f on the open unit discDwith the standard normalization f (0) = 0 = f ′(0) − 1 and satisfying  1 1 − α ( z(Dq f )(z) h(z) − α ) − 1 1 − q  ≤ 1 1 − q , (z ∈ D), where h ∈ Sq. This class is denoted by Kq(α), so called the class of q-close-to-convexfunctions of order α. In this paper, we study some geometric properties of this class. In addition, we consider the famous Bieberbach conjecture problem on coefficients for the class Kq(α). We also find some sufficient conditions for the function to be in Kq(α) for some particular choices of the functions h. Finally, we provide some applications on q-analogue of Gaussian hypergeometric function.

A certain class of q-close-to-convex functions of order α