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On a generalization of close-to-convex functions

Volume 113 / 2015

Swadesh Kumar Sahoo, Navneet Lal Sharma Annales Polonici Mathematici 113 (2015), 93-108 MSC: Primary 30C45; Secondary 30B10, 30C50, 30C55, 33B30, 33D15, 40A30, 47E05. DOI: 10.4064/ap113-1-6

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.

Authors

  • Swadesh Kumar SahooDepartment of Mathematics
    Indian Institute of Technology Indore
    Indore 452 017, India
    e-mail
  • Navneet Lal SharmaDepartment of Mathematics
    Indian Institute of Technology Indore
    Indore 452 017, India
    e-mail

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