On Subclass of Analytic Functions Defined by q-Analogue of Modified Tremblay Fractional Derivative Operator

Year 2024, Volume: 6 Issue: 2, 44 - 53

Abdullah Alsoboh , Waggas Galıb

https://doi.org/10.47086/pims.1535676

Abstract

In this research, by using the principle of quantum calculus, we introduce a modified fractional derivative operator $\mathcal{T}^{\xi,\digamma}_{q,\varsigma}$ of the analytic functions in the open unit disc $\diamondsuit=\{\varsigma:\varsigma\in\mathbb{C},|\varsigma|<1\}$. The operator $\mathcal{T}^{\xi,\digamma}_{q,\varsigma}$ can then be used to introduce a new subclass of analytic functions $\mathcal{D}\bigoplus(\vartheta,\digamma,d,\xi,\gamma;q)$. We present the necessary conditions for functions belonging to the subclass $\mathcal{D}\bigoplus(\vartheta,\digamma,d,\xi,\gamma;q) $.\\

Furthermore, we discuss a growth and distortion bounds, the convolution condition, and the radii of starlikeness. In addition, we present neighbourhoods problems involving the $\mathfrak{q}$-analogue of a modified Tremblay operator for functions in the introduced class $\mathcal{D}\bigoplus(\vartheta,\digamma,d,\xi,\gamma;q)$.

Keywords

Analytic functions, Starlikeness, Radii of starlikeness, Neighbourhoods Problems, Fractional Operator, q-calculus

Ethical Statement

This research is the original work of the authors and has not been published elsewhere. The authors confirm that this manuscript complies with the ethical standards of the journal and that no data fabrication, falsification, plagiarism, or inappropriate data manipulation occurred during the research.

Supporting Institution

Philadelphia University

References

• P.R. Agarwal, Certain fractional q-integrals and q-derivatives, Math. Proc. Camb. Philos. Soc. 66(2)(1969) 2365–370.
• I. Aldawish and M. Darus, tarlikeness of q-differential operator involving quantum calculus, The Kangwon-Kyungki Math. Soc. 22(4) (2020) 699–709.
• A. Alb Lupa¸s, and G. Oros, Fractional integral of a confluent hypergeometric function applied to defining a new class of analytic functions. Symmetry 14 (2022) 427.
• W. Al-Salam, A. Verma, A fractional Leibniz q-formula, Pacific Journal Of Mathematics 60 (1975) 1–9.
• W. Al-Salam, Some fractional q-integrals and q-derivatives, Proceedings Of The Edinburgh Mathematical Society 15 (1966) 135-140.
• W. Atshan, A. Battor, and N. Abbas, On a new subclass of univalent functions with positive coefficients defined by Ruscheweyh Derivative, Eur. Jour. Sci. Res. 140 (2016) 85–92.
• S. Elhaddad, M. Darus, On meromorphic functions defined by a new operator containing the Mittag–Leffler function. Symmetry, (11)(2) (2019), 210.
• Z. Esa, A. Kilicman, R. Ibrahim, M. Ismail and S. Husain, Application of modified complex tremblay operator. AIP Conference Proceedings 1739 (2016), 020059.
• B. Frasin and M. Darus, Subclass of analytic functions defined by q-derivative operator associated with Pascal distribution series. AIMS Math, 6 (2021) 5008-5019
• Exton, H. q-Hypergeometric functions and applications, E. Horwood Halsted Press, (1983).
• G. Gasper, and Rahman, M. Basic hypergeometric series, Cambridge university press (2011).
• A. Kilbas, H. Srivastava and J. Trujillo, Theory and applications of fractional differential equations. Cambridge university press (2011).
• M. Garg and L. Chanchlani, q-analogues of Saigo’s fractional calculus operators, Bull. Math. Anal. Appl. 3 (2011) 169–179.
• S. Isogawa, N. Kobachi and S. Hamada, A q-analogue of Riemann-Liouville fractional derivative, Res. Rep. Yatsushiro Nat. Coll. Tech. 29 (2007) 59–68.
• I. Podlubny, An introduction to fractional derivatives, fractional differential equations to methods of their solution and some of their applications, Math. Sci. Eng. 198 (1999), 0924-34008
• J. Prajapat, R. Raina and H. Srivastava, Some inclusion properties for certain subclasses of strongly starlike and strongly convex functions involving a family of fractional integral operators, Int. Tran. Spec. Fun. 18 (2007) 639–651.
• S. Purohit and R. Raina, Certain subclasses of analytic functions associated with fractional q-calculus operators, Math. Scand., 109, (2011) 55–70
• S. Purohit, and R. Yadav, On generalized fractional q-integral operators involving the qGauss hypergeometric function, Bull. Math. Anal. Appl. 2 (2010) 35–44.
• S. Ramadan and M. Darus, Univalence of an integral operator defined by generalized operators, Int. J. Math. Comput. Sci. 4 (2010) 1117–1119.
• S. Samko, Fractional integrals and derivatives: Theory And Applications, Gordon and Breach, (1993).
• R. Saxena, R. Yadav, S. Purohit and S. Kalla, Kober fractional q-integral operator of the basic analogue of the H-function, Rev. T´ec. Fac. Ing. Univ. Zulia. 28 (2005) 154-158.
• T. Seoudy and M. Aouf, Coefficient estimates of new classes of q-starlike and q-convex functions of complex order, J. Math. Inequal 10 (2016) 135-145.
• R. Tremblay, Une Contribution a la Theorie de la Derivee Fractionnaire. Ph. D. Thesis, (1974).