There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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

For q ∈ (0, 1) let the q-difference operator be defined as follows 

$$\partial _q f(z) = \frac{{f(qz) - f(z)}} {{z(q - 1)}} (z \in \mathbb{U}),$$

where $\mathbb{U}$ denotes the open unit disk in a complex plane. Making use of the above operator the extended Ruscheweyh differential operator Rqλf is defined. Applying Rqλf a subfamily of analytic functions is defined. Several interesting properties of a defined family of functions are investigated.

/pdf/some-class-of-analytic-functions-related-to-conic-domains-3ls909z0jy.pdf

Some class of analytic functions related to conic domains

In terms of the Hadmard product (or convolution), define the operator by , where the function f is analytic in the unit disk. The classes of k-uniformly convex and k-starlike functions , denoted by , respectively, were introduced recently (cf.[3] and [5]). The object of the present paper is to find conditions on the papameters a,b,c, and k, for which the linear operator maps the classes of starlike and univalent functions onto .

Linear operators associated with k-uniformly convex functions

Let $\fancyscript{S}_{e}^*$ denote the class of analytic functions $f$ in the open unit disk normalized by $f(0)=f'(0)-1=0$ and satisfying the condition $zf'(z)/f(z)\prec e^z$ for $|z|<1$. The structural formula, inclusion relations, coefficient estimates, growth and distortion results, subordination theorems and various radii constants for functions in the class $\fancyscript{S}_{e}^*$ are obtained. In addition, the sharp $\fancyscript{S}_{e}^*$-radii for functions belonging to several interesting classes are also determined.

On a Subclass of Strongly Starlike Functions Associated with Exponential Function

http://directory.umm.ac.id/Data%20Elmu/jurnal/J-a/Journal%20of%20Computational%20And%20Applied%20Mathematics/Vol105.Issue1-2.1999/2735.pdf

Conic regions and k -uniform convexity

We solve the problem of finding the largest domain D for which, under given ψ and q, the differential subordination ψ(p(z),zp′(z))∈D⇒p(z)≺q(z), where D and q(&#119984;) are regions bounded by conic sections, is satisfied. The shape of the domain D is described by the shape of q(&#119984;). Also, we find the best dominant of the differential subordination p(z)

/pdf/techniques-of-the-differential-subordination-for-domains-1vknkt0vm6.pdf

Stanisława Kanas

Papers

Conic regions and k -uniform convexity

Some class of analytic functions related to conic domains

Linear operators associated with k-uniformly convex functions

Techniques of the differential subordination for domains bounded by conic sections

An unified approach to the Fekete–Szegö problem