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.

Operators of Basic (or q-) Calculus and Fractional q-Calculus and Their Applications in Geometric Function Theory of Complex Analysis

Univalent functions, fractional calculus, and their applications , edited by H. M. Srivastava and S. Owa. Pp 404. £39·95. 1989. ISBN 0-7458-0701-1 (Ellis Horwood)

In this paper, the new generalized classes of (p, q)-starlike and (p, q)-convex functions are introduced by using the (p, q)-derivative operator. Also, the (p, q)-Bernardi integral operator for analytic function is defined in the open unit disc $$\mathbb {U}=\left\{ z\in \mathbb {C}:|z|<1\right\} $$. Our aim for these classes is to investigate the Fekete-Szego inequalities. Moreover, Some special cases of the established results are discussed. Further, certain applications of the main results are obtained by applying the (p, q)-Bernardi integral operator.

Fekete-Szegö inequality for classes of (p, q)-Starlike and (p, q)-convex functions

The core object of this paper is to define and study a new class of analytic functions using the Ruscheweyh q-differential operator. We also investigate a number of useful properties of this class such structural formula and coefficient estimates for functions. We consider also the Fekete–Szego problem in the class, we give some subordination results, and some other corollaries.

New Subclass of Analytic Functions in Conical Domain Associated with Ruscheweyh q-Differential Operator

The main purpose of this article is to find the upper bound of the third Hankel determinant for a family of q-starlike functions which are associated with the Ruscheweyh-type q-derivative operator. The work is motivated by several special cases and consequences of our main results, which are pointed out herein.

Upper Bound of the Third Hankel Determinant for a Subclass of q-Starlike Functions

The main purpose of this investigation is to find several coefficient inequalities and a sufficient condition for $q$-starlike functions which are associated with the Janowski functions. Relevant connections of the results presented in this paper with those in a number of other related works on this subject are also pointed out.

Coefficient inequalities for $q$-starlike functions associated with the Janowski functions

By using a certain general conic domain as well as the quantum (or q-) calculus, here we define and investigate a new subclass of normalized analytic and starlike functions in the open unit disk U . In particular, we find the Hankel determinant and the Toeplitz matrices for this newly-defined class of analytic q-starlike functions. We also highlight some known consequences of our main results.

Hankel and Toeplitz Determinants for a Subclass of q-Starlike Functions Associated with a General Conic Domain

By making use of the concept of basic (or q-) calculus, various families of q-extensions of starlike functions of order α in the open unit disk U were introduced and studied from many different viewpoints and perspectives. In this paper, we first investigate the relationship between various known classes of q-starlike functions that are associated with the Janowski functions. We then introduce and study a new subclass of q-starlike functions that involves the Janowski functions. We also derive several properties of such families of q-starlike functions with negative coefficients including (for example) distortion theorems.

https://www.mdpi.com/2073-8994/11/2/292/pdf

Some General Classes of q-Starlike Functions Associated with the Janowski Functions

In this paper, the authors introduce a new subclass of meromorphic q-starlike functions which are associated with the Janowski functions. A characterization of meromorphically q-starlike functions associated with the Janowski functions has been obtained when the coefficients in their Laurent series expansion about the origin are all positive. This leads to a study of coefficient estimates, distortion theorems, partial sums, and the radius of starlikeness estimates for this class. It is seen that the class considered demonstrates, in some respects, properties analogous to those possessed by the corresponding class of univalent analytic functions with negative coefficients.

/pdf/a-certain-subclass-of-meromorphically-q-starlike-functions-3ly7egbat0.pdf

Qazi Zahoor Ahmad

Papers

Coefficient inequalities for $q$-starlike functions associated with the Janowski functions

Hankel and Toeplitz Determinants for a Subclass of q-Starlike Functions Associated with a General Conic Domain

Some General Classes of q-Starlike Functions Associated with the Janowski Functions

Upper Bound of the Third Hankel Determinant for a Subclass of q-Starlike Functions

A certain subclass of meromorphically q-starlike functions associated with the Janowski functions