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Fractal Fract 2019, 3(1), 10; https://doi.org/10.3390/fractalfract3010010
On Analytic Functions Involving the q-Ruscheweyeh Derivative
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.
Keywords:univalent; q-starlike; q-difference operator; subordination
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 writewhere ≺ 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
For letso that
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:andprovided that the series converges.
It can easily be seen that for andwhere
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 aswhere the series is absolutely convergent in
The q-Ruscheweyh differential operator of order and for f given by (1) is defined as
Equation (5) can be written as
Since it follows that
Throughout this paper, it is assumed that and unless otherwise stated.
2. Main Results
In this section, some new classes of analytic functions involving the q-Ruscheweyh derivative are introduced and some new results are derived.
Let Then, f is said to belong to the class ifwhere is defined by (2) on the set
We note that as the disc becomes the right half plane and the class reduces to
It can be seen from (7) that the transformation maps onto the circle with center and the radius which can be written as
Now, with and some computation, (8) yields
Taking the q-integral on both sides of (9) together with some simplifications, we obtain the following result for the class
Since we obtain the well known distortion result for as
Let Then, f is said to belong to the class , if and only ifand is defined by (5).
As a special case, we have if and only if ,
The following identity can easily be obtained from (5).
If , thenwhich is the well known identity of the Ruscheweyeh derivative operator
With a similar argument used in Reference , it can easily be shown that
Setwhere is analytic in E with
We can show that
Clearly, is analytic in E and We can show S
Suppose on the contrary that there exists a such that Since for and
Now, from (15), it follows that
At we have
Using q-Jacks’s Lemma given in Reference , we have
Let and let be given by (1). Then,and is a constant depending only on
By Cauchy theorem, for and Cauchy–Schwartz inequality, we have
If and then and
Using Theorem 1 and the subordination principle, we obtainwhere is a constant.
This completes the proof. □
As a special case, if then and
Letwhere is given by (5) and Then,where is a constant which depends only on
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.
- Abu Arqub, O. Solutions of time-fractional Tricomi and Keldysh equations of Dirichlet functions types in Hilbert space. Numer. Methods Part. Differ. Equ. 2018, 117, 1759–1780. [Google Scholar] [CrossRef]
- Noor, K.I. On some classes of analytic functions associated with q-Ruscheweyh differential operator. FACTA Universitatis Ser. Math. Inform. 2018, 33, 531–538. [Google Scholar]
- Noor, K.I.; Arif, M. On some applications of the Ruscheweyh derivative. Comput. Math. Appl. 2011, 62, 4727–4732. [Google Scholar] [CrossRef]
- Arif, M.; Srivastava, H.M.; Umar, S. Some applications of a q-analogue of the Ruscheweyh type operator for multivalent functions. RACSAM 2018. [Google Scholar] [CrossRef]
- Hristov, J. Approximate solutions to fractional subdiffusion equations. Eur. Phys. J. Spec. Top. 2018, 193, 229–243. [Google Scholar] [CrossRef]
- Miller, K.S.; Ross, B. An Introduction to the Fractional Calculus and Fractional Differential Equations; Wiley-Interscience Publication: New York, NY, USA, 1993. [Google Scholar]
- Dos Santos, M. Non-Gaussian distributions to random walk in the context of memory kernels. Fract. Fract. 2018, 3, 20. [Google Scholar] [CrossRef]
- Ademogullari, K.; Kahramaner, Y. q-harmonic mappings for which analytic part is q-convex function. Nonlinear Anal. Differ. Equ. 2016, 4, 283–293. [Google Scholar] [CrossRef]
- Jackson, F.H. q-Differential equations. Am. J. Math. 1910, 32, 305–314. [Google Scholar] [CrossRef]
- Jackson, F.H. On q-definite integral. Quart. J. Pure Appl. Math. 1910, 41, 193–203. [Google Scholar]
- Ismail, M.E.H.; Merkes, E.; Styer, D. A generalization of starlike functions. Complex Var. 1990, 14, 77–84. [Google Scholar] [CrossRef]
- Ernest, T. The History of q-Calculus and a New Method. Licentia Dissertation, Uppsala University, Uppsala, Sweden, 2001. [Google Scholar]
- Kac, V.; Cheung, P. Quantum Calculus; Springer: New York, NY, USA, 2002. [Google Scholar]
- Abu Arqub, O.; Al-Smadi, M. Atangana–Baleanu fractional approach to the solutions of Bagley–Torvik and Painlevé equations in Hilbert space. Chaos Solitons Fractals 2018, 117, 161–167. [Google Scholar] [CrossRef]
- Abu Arqub, O.; Maayah, B. Numerical solutions of integrodifferential equations of Fredholm operator type in the sense of the Atangana–Baleanu fractional operator. Chaos Solitons Fractals 2018, 117, 117–124. [Google Scholar] [CrossRef]
- Abu Arqub, O.; Al-Smadi, M. Numerical algorithm for solving time-fractional partial integrodifferential equations subject to initial and Dirichlet boundary conditions. Numer. Methods Part. Differ. Equ. 2018, 34, 1577–1597. [Google Scholar] [CrossRef]
- Noor, K.I.; Riaz, S.; Noor, M.A. On q-Bernardi Integral Operator. TWMS J. Pure Appl. Math. 2017, 8, 3–11. [Google Scholar]
- Noor, K.I. On generalized q-Bazilevic functions. J. Adv. Math. Stud. 2017, 10, 418–424. [Google Scholar]
- Noor, K.I. Some classes of q-alpha starlike and related analytic functions. J. Math. Anal. 2017, 8, 24–33. [Google Scholar]
- Noor, K.I.; Riaz, S. Generalized q-starlike functions. Stu. Sci. Math. Hung. 2017, 54, 1–14. [Google Scholar] [CrossRef]
- Noor, K.I.; Riaz, S. On certain classes of analytic functions involving q-difference operator. Acta Univ. Sapientiae Math. 2018, 10, 178–188. [Google Scholar]
- Kanas, S.; Raducanu, D. Some classes of analytic functions related to Conic domains. Math. Slovaca 2014, 64, 1183–1196. [Google Scholar] [CrossRef]
- Ucar, H.E.O. Coefficient inequality for q-starlike functions. Appl. Math. Comput. 2016, 276, 122–126. [Google Scholar]
- Bernardi, S.D. Convex and starlike univalent functions. Trans. Am. Math. Soc. 1969, 135, 429–446. [Google Scholar] [CrossRef]
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