On a generalization of close-to-convex functions

Swadesh Kumar Sahoo¹ ; Navneet Lal Sharma¹

¹ Department of Mathematics Indian Institute of Technology Indore Indore 452 017, India

Annales Polonici Mathematici, Tome 113 (2015) no. 1, pp. 93-108

Résumé

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.

Détail

DOI : 10.4064/ap113-1-6

Keywords: paper ismail complex variables theory appl motivates study generalization close to convex functions means q analog difference operator acting analytic functions unit disk mathbb mathbb term q close to convex functions q analog close to convex functions obtain conditions coefficients power series functions analytic unit disk which ensure generate functions q close to convex family result certain dilogarithm functions contained family secondly study bieberbach problem coefficients analytic q close to convex functions produces several power series analytic functions convergent basic hypergeometric functions