Department of Mathematics, COMSATS, University Islamabad, Islamabad 44000, Pakistan
Received: 20 February 2019 / Accepted: 8 March 2019 / Published: 10 March 2019
In this paper, we use concepts of q-calculus to introduce a certain type of q-difference operator, and using it define some subclasses of analytic functions. Inclusion relations, coefficient result, and some other interesting properties of these classes are studied.
Let A denote the class of functions f which are analytic in the open unit disc and are of the form
One-to-one analytic functions in this class are usually called univalent. A function is said to be starlike of order in E if it satisfies the condition
We denote this class by In particular, for we have the well-known class of starlike functions. The class consisting of convex functions of order can be defined by the relation
Let If there exists a Schwartz function analytic in E with such that for all such that then we say that is subordinate to and write
where ≺ denotes subordination.
Let f and g be analytic in E with and Then, the convolution ∗ (or Hadamard product) of f and g is defined as
The operator is called the Ruscheweyh derivative of order see Reference . For the applications of the Ruscheweyh differential operator in geometric function theory, see References [2,3,4].
In this paper, we generalize the operator by using q-calculus concepts. Recently, q-calculus has attracted the attention of many researchers in the field of geometric function theory. q-Derivatives and q-integrals play an important and significant role in the study of quantum groups and q-deformed super-algebras, the study of fractal and multi-fractal measures, and in chaotic dynamical systems. The name q-calculus also appears in other contexts; see References [5,6,7,8,9,10,11,12,13]. The most sophisticated tool that derives functions in non-integer order is the well-known fractional calculus; see References [1,12,13,14,15,16]. One can find numerous applications of the q-operator in real-world problems as well as in problems defined on complex plains.
Ismail et al.  generalized the class with the concept of q-derivative and called it of q-starlike functions. Here, we give some basic definitions and results of q-calculus which we shall use in our results. For more details, see References [12,13,17,18,19,20,21,22].
If is fixed, then a subset B of is called q-geometric, if whenever and B contains all geometric sequences Jackson [9,10] defined q-derivative and q-integral of f on the set B as follows:
provided that the series converges.
It can easily be seen that for and
For any non-negative integer m, the q-number shift factorial is defined by
Furthermore, the q-generalized Pochhamer symbol for is given as
Let the function F be defined as
where the series is absolutely convergent in
The q-Ruscheweyh differential operator of order and for f given by (1) is defined as
The proof follows easily by using Lemma 1 and the definition that
As a special case, we observe that and we have
For it yields the result
In this paper, we have used q-calculus to define and study some new sub-classes of analytic functions involving the Ruscheweyh derivative.. Some interesting inclusion and subordination properties of these new classes have been derived. Some special cases have been discussed as applications of our main results. Applications of the q-Ruscheweyh differential operator in the real world will be an interesting and encouraging future study for researchers.
The author would like to the thank the Rector, COMSATS University Islamabad, Pakistan, for providing excellent research and academic environments. The author is grateful to the referees for their valuable suggestions and comments.
Conflicts of Interest
The author declares no conflict of interest.
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