## Abstract

Basic (or *q*-) series and basic (or *q*-) polynomials, especially the basic (or *q*-) hypergeometric functions and basic (or *q*-) hypergeometric polynomials, are known to have widespread applications, particularly in several areas of number theory and combinatorial analysis such as (for example) the theory of partitions. Our usages here, in this *survey-cum-expository* article, of the *q*-calculus and the fractional *q*-calculus in geometric function theory of complex analysis are believed to encourage and motivate significant further developments on these and other related topics. By applying a fractional *q*-calculus operator, we define the subclasses \({\mathcal{S}}_{n}^{\alpha }(\lambda ,\beta ,b,q)\) and \({\mathcal{G}}_{n}^{\alpha }(\lambda ,\beta ,b,q)\) of normalized analytic functions with complex order and negative coefficients.
Among the results investigated for each of these function classes, we derive their associated coefficient estimates, radii of close-to-convexity, starlikeness and convexity, extreme points and growth and distortion theorems. Our investigation here is motivated essentially by the fact that basic (or *q*-) series and basic (or *q*-) polynomials, especially the basic (or *q*-) hypergeometric functions and basic (or *q*-) hypergeometric polynomials, are applicable particularly in several areas of number theory such as the theory of partitions. In fact, basic (or *q*-) hypergeometric functions are useful also in a wide variety of fields including, for example, combinatorial analysis, finite vector spaces, lie theory, particle physics, nonlinear electric circuit theory, mechanical engineering, theory of heat conduction, quantum mechanics, cosmology and statistics (see also (Srivastava and Karlsson in Multiple Gaussian hypergeometric series. pp 350–351, 1985) and the references cited thereon). In the last section on conclusion, we choose to point out the fact that the results for the *q*-analogues, which we consider in this article for \(0< q < 1\), can easily (and possibly trivially) be translated into the corresponding results for the (*p*, *q*)-analogues (with \(0< q < p \leqq 1\)) by applying some obvious parametric and argument variations, the additional parameter *p* being redundant. Several other families of such extensively- and widely-investigated linear convolution operators as (for example) the Dziok–Srivastava, Srivastava–Wright and Srivastava–Attiya linear convolution operators, together with their extended and generalized versions, are also briefly considered.

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Srivastava, H.M. Operators of Basic (or *q*-) Calculus and Fractional *q*-Calculus and Their Applications in Geometric Function Theory of Complex Analysis.
*Iran J Sci Technol Trans Sci* **44, **327–344 (2020). https://doi.org/10.1007/s40995-019-00815-0

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### Keywords

- Analytic functions
- Fractional
*q*-calculus operators *q*-Gamma functions- Starlike and
*q*-starlike functions of complex order - Convex and
*q*-convex functions of complex order - Close-to-convex and
*q*-close-to-convex functions of complex order - Coefficient estimates
- Radii of close-to-convexity, starlikeness and convexity
- Extreme points
- Growth and distortion theorems
- The classical
*q*-analysis versus the so-called (*p*,*q*)-analysis - Dziok–Srivastava, Srivastava–Attiya and Srivastava–Wright linear convolution operators

### Mathematics Subject Classification

- Primary 11M35
- 26A33
- 30C45
- 33B15
- Secondary 11M06
- 11B68
- 33D05
- 33E20
- 33E30