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 𝔻 = {z ∈ 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ap113-1-6,
author = {Swadesh Kumar Sahoo and Navneet Lal Sharma},
title = {On a generalization of close-to-convex functions},
journal = {Annales Polonici Mathematici},
volume = {113},
year = {2015},
pages = {93-108},
zbl = {1308.30020},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap113-1-6}
}
Swadesh Kumar Sahoo; Navneet Lal Sharma. On a generalization of close-to-convex functions. Annales Polonici Mathematici, Tome 113 (2015) pp. 93-108. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap113-1-6/