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The word “symmetry” is a Greek word that originated from “symmetria”. It means an agreement in dimensions, due proportion, and arrangement; however, in complex analysis, it means objects remaining invariant under some transformation. This idea has now been recently used in geometric function theory to modify the earlier classical q-derivative introduced by Ismail et al. due to its better convergence properties. Consequently, we introduce a new class of analytic functions by using the notion of q-symmetric derivative. The investigation in this paper obtains a number of the latest important results in q-theory, including coefficient inequalities and convolution characterization of q-symmetric starlike functions related to Janowski mappings.
Let be the class of analytic functions having the series form
Let denote the subclasses of consisting of functions that are univalent in ℧. We say is subordinate to (written as or ) if there exists a Schwarz function such that for all . For with and , the convolution of f and g depicted by is defined as
Let denote the class of all functions such that the following subordination condition is satisfied:
If we choose and in (2), then and , respectively . In particular, if , the class reduces to the usual class of functions with positive real part, and and of starlike and convex functions, respectively.
q-calculus is a significant concept in modern mathematics. It also plays a crucial role in many fields of physics such as cosmic strings and black holes, nuclear and high energy physics . This idea of q-calculus was developed by Jackson  and its calculus is based on q-derivative
We observed that several results in the area of q-theory are analogs of the important results from the classical analysis.
In geometric function theory (GFT), Ismail et al.  first utilized the q-derivative to define the class of starlike functions. As a result, numerous articles (which contain new ideas or nice extensions of the classical classes in GFT) are scattered in the literature. We refer the reader to [6,7,8,9,10,11] and the references cited therein, for the most recent work; therefore, the generalization of q-calculus popped up in different subjects, such as complex analysis, hypergeometric series, statistics and particle physics. Alb Lupaş  used the techniques of differential subordination to study the geometric properties of q-Sălăgean differential operator. Altintaş and Mustafa  introduced new classes of analytic functions defined by q-operator and gave the necessary condition for analytic functions to be members of those classes. In addition, they established the growth and distortion results related with these families of functions. Closely related to the classes of Altintaş and Mustafa, Orhan et al.  studied the Fekete–Szegö problem connected to a new class of analytic functions.
However, in the “Survey-cum-expository” by Srivastava , it was noted that the so-called -calculus extension is a rather trivial and inconsequential variation of the classical q-calculus, the additional parameter p being redundant.
For a fixed and , the q-symmetric derivative of a function at a point is defined by
The q-symmetric quantum calculus has been resourceful in many areas of study; for instance, in quantum mechanics. It was noted in  that the q-symmetric derivative has, in general, better convergence properties than the classical q-derivative.
Recently, this concept of the derivative has been used to introduce and study different classes of univalent functions. In this direction, Kanas et al. , using the notion of the symmetric operator of q-derivative, defined and studied a new family of univalent functions in a conic region. Khan et al. [16,17] slightly modified this Kanas class and investigated certain properties associated with the class, which include structural formula, necessary and sufficient conditions, coefficient estimates, Fekete–Szegö problem, distortion inequalities, closure theorem and subordination results. It is worthy of note that results presented by Khan et al. in [16,17] have no significant difference. Moreover, Seoudy  introduced certain classes of symmetric q-starlike and symmetric q-convex functions. For these classes, he obtained convolution properties and coefficient inequalities. Zhang et al.  initiated symmetric Salagean q-differential operator and then used it to introduce the class of harmonic univalent functions. Then, they examined many interesting properties associated with the defined class. Furthermore, very recently, Khan et al.  extended the notion of q-symmetric derivative to multivalent functions. They introduced multivalent q-symmetric starlike functions and obtained its geometric characterizations.
Motivated by these current developments, we initiate the class of q-symmetric starlike functions of the Janowski type and examine many coefficient inequalities and sufficient conditions for this class. In addition, a convolution property for it is established.
Next, we present some fundamental preliminaries which are necessary for our findings.
(). Let . Then, the symmetric q-number denoted by is defined as
and the symmetric q-derivative of a function in ℧ is given by
We note that the symmetric q- number is not reducible to the classical q-number. It is cleared from the above definition that for given by (1), we have
Let , we have the following rules for q-symmetric difference operator.
(). Let be q-symmetric differentiable and . Then
One way to extend the class is to assume that the function
Then, the appropriate definition of the corresponding class is given as:
Let , and . Then if and only if
Equivalently, if and only if
For and , then reduces to .
For and as , then is equivalent to the usual class of starlike functions.
(Subordinating Factor Sequence). A sequence of complex number is called a subordinating factor sequence if, whenever of the form (1) is analytic, univalent and convex in ℧, we have the subornation given by
The following results are required for our findings.
Conversely, since , then in ℧. Therefore, the function is analytic in ℧ with . In the first part of the proof, we observe that (20) and
are equivalent. Let
Thus, the connected part of contains the simply connected domain ; therefore, the univalence of the function in ℧ and the fact that affirm that in ℧. Hence, . □
In this findings, we introduced the class of analytic functions by using the notion of q-symmetric derivative, and obtained coefficient related results. Furthermore, some convolution characterization associated with were presented. The consequences of our investigation include known and new results.
It is interesting to note that this presented work could be investigated under the context of multivalent functions and some geometric characterizations such as the Fekete–Szegö inequality, Hankel determinant, growth and distortion problems could be explored. In addition, using the theory of differential subordination, Sandwich-type results could be examined for this present class of functions. For more details about the suggested work, one may go through [28,29]. Overall, the results presented here could represent a starting point for full investigations into the study of Janowski functions in the framework of q-symmetric calculus.
Conceptualization, A.S.; Formal analysis, A.S.; Investigation, A.S., I.A.-S., S.N.M., J.G. and N.A; Methodology, A.S., I.A.-S., S.N.M., J.G. and N.A; Project administration, A.S., I.A.-S., S.N.M. and J.G.; Validation, A.S., I.A.-S., S.N.M. and J.G.; Writing—original draft, A.S.; Writing— review and editing, A.S., I.A.-S., S.N.M., J.G. and N.A. All authors have read and agreed to the published version of the manuscript.
This research received no external funding.
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
The first author is indebted to his teachers, Khalida Inayat Noor (at COMSATS University Islamabad, Pakistan) and K.O Babalola (at University of Ilorin, Ilorin, Nigeria), for their active roles in his academic development.
Conflicts of Interest
The authors declare no conflict of interest.
Miller, S.S.; Mocanu, P.T. Differential Subordinations. In Theory and Applications; Marcel Dekker Inc.: New York, NY, USA, 2000. [Google Scholar]
Janowski, W. Some extremal problems for certain families of analytic functions I. Ann. Pol. Math.1973, 28, 297–326. [Google Scholar] [CrossRef]
Brito da Cruz, A.M.C.; Martins, N. The q-symmetric variational calculus. Comput. Math. Appl.2012, 64, 2241–2250. [Google Scholar] [CrossRef]
Ismail, M.E.H.; Merkes, E.; Styer, D. A generalization of starlike functions. Complex Var. Theory Appl. Int. J.1990, 14, 77–84. [Google Scholar] [CrossRef]
Olatunji, S.O. Fekete–Szegö inequalities on certain subclasses of analytic functions defined by λ-pseudo-q-difference operator associated with s-sigmoid function. Bol. Soc. Mat. Mex.2022, 28, 1–15. [Google Scholar] [CrossRef]
Piejko, K.; Sokół, J.; Trabka-Wieclaw, K. On q-calculus and starlike functions. Iran. J. Sci. Technol. Trans. A Sci.2019, 43, 2879–2883. [Google Scholar] [CrossRef]
Piejko, K.; Sokół, J. On convolution and q-calculus. Bol. Soc. Mat. Mex.2020, 26, 349–359. [Google Scholar] [CrossRef]
Saliu, A.; Noor, K.I.; Hussain, S.; Darus, M. On Quantum Differential Subordination Related with Certain Family of Analytic Functions. J. Math.2020, 2020, 6675732. [Google Scholar] [CrossRef]
Srivastava, H.M. Operators of basic (or q-) calculus and fractional q-calculus and their applications in geometric function theory of complex analysis. Iran. J. Sci. Technol. Trans. A Sci.2020, 44, 327–344. [Google Scholar] [CrossRef]
Verma, S.; Kumar, R.; Sokół, J. A conjecture on Marx-Strohhäcker type inclusion relation between q-convex and q-starlike functions. Bull. Sci. Math.2020, 174, 103088. [Google Scholar] [CrossRef]
Alb Lupaş, A. Subordination Results on the q-Analogue of the Sălăgean Differential Operator. Symmetry2022, 14, 1744. [Google Scholar] [CrossRef]
Altintaş, O.; Mustafa, N. Distortion bound andgrowth theorems for a subclass of analytic functionsdefined by q-derivative. Turk. J. Math.2022, 46, 2096–2108. [Google Scholar] [CrossRef]
Orhan, H.; Porwal, S.; Magesh, N. The Fekete-Szegö problem for a generalized class ofanalytic functions of complex order associated with q-calculus. Palest. J. Math.2022, 11, 39–47. [Google Scholar]
Kanas, S.; Altinkaya, Ş.; Yalçin, S. Subclass of k-Uniformly Starlike Functions Defined by the Symmetric q-Derivative Operator. Ukr. Math. J.2012, 70, 1727–1740. [Google Scholar] [CrossRef]
Khan, S.; Khan, N.; Hussain, A.; Araci, S.; Khan, B.; Al-Sulami, H.H. Applications of Symmetric Conic Domains to a Subclass of q-Starlike Functions. Symmetry2022, 14, 803. [Google Scholar] [CrossRef]
Khan, S.; Hussain, S.; Naeem, M.; Darus, M.; Rasheed, A. A subclass of q-starlike functions defined by using a symmetric q-derivative operator and related with generalized symmetric conic domains. Mathematics2022, 9, 917. [Google Scholar] [CrossRef]
Seoudy, T.M. Convolution Results and Fekete–Szegö Inequalities for Certain Classes of Symmetric-Starlike and Symmetric-Convex Functions. J. Math.2022, 2022, 8203921. [Google Scholar] [CrossRef]
Zhang, C.; Khan, S.; Hussain, A.; Khan, N.; Hussain, S.; Khan, N. Applications of q-difference symmetric operator in harmonic univalent functions. AIMS Math.2022, 7, 667–680. [Google Scholar] [CrossRef]
Khan, M.F.; Goswami, A.; Khan, S. Certain New Subclass of Multivalent Q-Starlike Functions Associated with Q-Symmetric Calculus. Fractal Fract.2022, 6, 367. [Google Scholar] [CrossRef]
Kamel, B.; Yosr, S. On some symmetric q-special functions. Matematiche2013, 68, 107–122. [Google Scholar]
Ma, W.C.; Minda, D. A unified treatment of some special classes of univalent functions. In Proceedings of the International Conference on Complex Analytic at the Nankai Institute of Mathematics, Tianjin, China, 19–23 June 1992; pp. 157–169. [Google Scholar]
Wilf, H.S. Subordinating factor sequences for convex maps of the unit circle. Proc. Am. Math. Soc.1961, 12, 689–693. [Google Scholar] [CrossRef]
Clunie, J.; Keogh, F.R. On starlike and convex schlicht functions. J. Lond. Math. Soc.1960, 1, 229–233. [Google Scholar] [CrossRef]
Nehari, Z. Conformal Mapping; McGraw-Hill: New York, NY, USA, 1952. [Google Scholar]
Holland, F.; Thomas, D.K. The area Theorem for starlike functions. J. Lond. Math. Soc.1969, 2, 127–134. [Google Scholar] [CrossRef]
Hayami, T.; Owa, S. Hankel determinant for p-valently starlike and convex functions of order α. Gen. Math.2009, 4, 29–44. [Google Scholar]
Hadi, S.H.; Darus, M. Differentialsubordination and superordination of a q-derivativeoperator connected with the q-exponential function. Int. J. Nonlinear Anal. Appl.2022, 13, 2795–2806. [Google Scholar] [CrossRef]
Khan, S.; Hussain, S.; Darus, M. Certainsubclasses of meromorphic multivalent q-starlike and q-convex functions. Math. Slovaca2022, 72, 635–646. [Google Scholar] [CrossRef]
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