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

Let $$\Delta $$
 be the open unit disk in the complex plane and $$\mathcal {A}$$
 be the class of normalized analytic functions in $$\Delta $$
 . In this paper, we introduce and study the class $$\begin{aligned} \mathcal {BS}(\alpha ):=\left\{ f\in \mathcal {A}: \left( \frac{zf'(z)}{f(z)}-1\right) \prec \frac{z}{1-\alpha z^2}, \, z\in \Delta \right\} , \end{aligned}$$
 where $$0\le \alpha \le 1$$
 and $$\prec $$
 is the subordination relation. Some properties of this class like differential subordination, coefficient estimates and Fekete–Szego inequality associated with the k-th root transform are considered.

On Booth lemniscate and starlike functions

In this paper, we introduce and investigate two new classes of meromorphic multivalent functions. Such results as subordination properties, coefficient inequalities, convolution properties and integral representations are proved. Several sufficient conditions for meromorphic multivalent starlikeness and convexity are also derived.

Certain classes of meromorphic multivalent functions

We obtain several inclusions between the class of functions with positive real part and the class of starlike univalent functions associated with the Booth lemniscate. These results are proved by applying the well-known theory of differential subordination developed by Miller and Mocanu and these inclusions give sufficient conditions for normalized analytic functions to belong to some subclasses of Ma–Minda starlike functions. In addition, by proving an associated technical lemma, we compute various radii constants such as the radius of starlikeness, radius of convexity, radius of starlikeness associated with the lemniscate of Bernoulli, and other radius estimates for functions in the class of functions associated with the Booth lemniscate. The results obtained are sharp.

https://journals.tubitak.gov.tr/math/issues/mat-18-42-3/mat-42-3-51-1706-49.pdf

Differential subordination and radius estimates for starlike functions associated with the Booth lemniscate

Let &#x1d49c; p (p∈ℕ:={1, 2, 3, …}) be the class of functions f given by which are analytic in the open unit disk &#x1d54c;. Making use of the method of differential subordinations, we obtain some sufficient conditions for f(z)∈&#x1d49c; p involving the Dziok–Srivastava operator to satisfy certain subordination properties. The results presented here are shown not only to generalize the main results in several earlier investigations, but also to give rise to many other new results.

Subordinations for multivalent analytic functions associated with the Dziok–Srivastava operator

Subclasses of meromorphic functions associated with the Cho-Kwon-Srivastava operator

On the Dziok–Srivastava operator under multivalent analytic functions

We consider the class $${\mathcal {S}}^*(\zeta ,\alpha )$$, $$0\le \alpha \alpha ,\quad |z|<1, \end{aligned}$$where $$\mathrm{d}_{\zeta }f$$ is the convolution operator $$\begin{aligned} \mathrm{d}_\zeta f(z) = \frac{1}{z}\left\{ f(z)*\frac{z}{(1-\zeta z)(1-z)}\right\} , \end{aligned}$$where $$\zeta $$ is complex, $$|\zeta |\le 1$$. For $$\zeta =1$$ the operator becomes the derivative $$f'$$, while for real $$\zeta =q$$, $$0<q<1$$, we obtain the Jackson q-derivative $$\mathrm{d}_qf$$.

/pdf/on-q-calculus-and-starlike-functions-3a3j7i6s45.pdf

On q-Calculus and Starlike Functions

In this paper we present some new applications of convolution and subordination in geometric function theory. The paper deals with several ideas and techniques used in this topic. Besides being an application to those results, it provides interesting corollaries concerning special functions, regions and curves.

/pdf/hadamard-product-of-analytic-functions-and-some-special-vj3mx5ubfr.pdf

Hadamard product of analytic functions and some special regions and curves

We consider the convolution operator $$\begin{aligned} \mathrm{d}_\zeta f(z) = \frac{1}{z}\left\{ f(z)*\frac{z}{(1-\zeta z)(1-z)}\right\} \end{aligned}$$on the class of analytic functions $$f(z)=z+a_2z^2+\cdots $$, $$|z|<1$$, in the complex plane, where $$\zeta $$ is complex, $$|\zeta |\le 1$$. For $$\zeta =1$$, the operator becomes the derivative $$f'(z)$$, while for real $$\zeta =q$$, $$0<q<1$$ we obtain the Jackson’s q-derivative $$\mathrm{d}_qf(z)$$.

/pdf/on-convolution-and-q-calculus-1wx7dgevzc.pdf

Krzysztof Piejko

Papers

Subclasses of meromorphic functions associated with the Cho-Kwon-Srivastava operator

On the Dziok–Srivastava operator under multivalent analytic functions

On q-Calculus and Starlike Functions

Hadamard product of analytic functions and some special regions and curves

On convolution and q-calculus