## On a generalization of close-to-convex functions

### Volume 113 / 2015

#### Abstract

The paper of M. Ismail et al. [Complex Variables Theory Appl. 14 (1990), 77–84] motivates the study of a generalization of close-to-convex functions by means of a $q$-analog of the difference operator acting on analytic functions in the unit disk $\mathbb {D}=\{z\in \mathbb {C}:|z|<1\}$. We use the term *$q$-close-to-convex functions* for the $q$-analog of close-to-convex functions. We obtain conditions on the coefficients of power series of functions analytic in the unit disk which ensure that they generate functions in the $q$-close-to-convex family. As a result we find certain dilogarithm functions that are contained in this family. Secondly, we also study the Bieberbach problem for coefficients of analytic $q$-close-to-convex functions. This produces several power series of analytic functions convergent to basic hypergeometric functions.