On New Subclasses of Analytic Functions Involving (p,q)-Derivatives

N Shilpa

doi:10.17485/IJST/v15i43.905

Article

On New Subclasses of Analytic Functions Involving (p,q)-Derivatives

VIEWS 708
PDF 124

Indian Journal of Science and Technology

DOI: 10.17485/IJST/v15i43.905

Year: 2022, Volume: 15, Issue: 43, Pages: 2336-2341

Original Article

On New Subclasses of Analytic Functions Involving (p,q)-Derivatives

N Shilpa^1*

¹Assistant Professor, PG Department of Mathematics, JSS College of Arts Commerce and Science, Mysuru, 570025, Karnataka, India

*Corresponding Author
Email: [email protected]

Received Date:23 May 2021, Accepted Date:11 February 2022, Published Date:18 November 2022

This work is licensed under a Creative Commons Attribution 4.0 International License.

Abstract

Objective: The objectives of the present study are to introduce some new subclasses of analytic functions involving (p,q)-derivatives by using subordination. We derive Fekete-Szegö inequalities for the functions belonging to the new subclasses. Method: Using the concept of (p,q)-derivative of a function and the subordination principle between analytic functions we introduce and study new subclasses. Findings: The Fekete-Szegö problem may be considered as one of the most important results about univalent functions. It was introduced by Fekete-Szegö in 1933. Coefficient estimates for the second and third coefficients of functions belonging to class of analytic functions with specific geometric properties were obtained. We obtain the Fekete-Szegö inequalities for functions belonging to the new subclasses. Moreover, some special cases of the established results are discussed. Novelty: The results of the paper are new and significantly contribute to the existing literature on the topic. Keywords: Analytic functions; Subordination; q-calculus; Fekete-Szegö inequalities; (p; q)-derivative operator

References

Jackson F. On q-definite integrals. Quarterly Journal of Pure and Applied Mathematics. 1910;41(15):193–203.
Jackson F. q-Difference Equations. American Journal of Mathematics. 1910;32(4):305–314. Available from: https://doi.org/10.2307/2370183
Frasin BA, Darus M. On the Fekete-Szegö problem. International Journal of Mathematics and Mathematical Sciences. 2000;24(9):577–581. Available from: https://doi.org/10.1155/S0161171200005111
Frasin BA, Darus M. Subclass of analytic functions defined by $ q $-derivative operator associated with Pascal distribution series. AIMS Mathematics. 2021;6(5):5008–5019. Available from: https://doi:10.3934/math.2021295
Frasin BA, Murugusundaramoorthy G. A subordination results for a class of analytic functions defined by Q differential operator. Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica. 2020;19(1):53–64. Available from: https:// doi: 10.2478/aupcsm-2020-0005
Aldweby MH, Darus. On Fekete-Szegö problems for certain subclasses defined by q-derivative. Journal of Function Spaces. 2017;p. 1–5. Available from: https:// doi.org/10.1155/2017/7156738
Alessa N, Venkateswarlu B, Loganathan K, Karthik TS, Reddy T, Sujatha P, et al. Certain class of analytic functions connected with Analogue of the Bessel function. Journal of Mathematics. 2021;9. Available from: https://doi.org/10.1155/2021/5587886
Srivastava R, Zayed H. Subclasses of analytic functions of complex order defined by q-derivative operator. Studia Universitatis Babes-Bolyai Mathematica. 2019;64(1):71–80. Available from: https://doi:10.24193/submath.2019.1.07
Mahmood S, Sokół J. New Subclass of Analytic Functions in Conical Domain Associated with Ruscheweyh q-Differential Operator. Results in Mathematics. 2017;71(3-4):1345–1357. Available from: https://doi: 10.1007/s00025-016-0592-1
Srivastava HM. Operators of Basic (or q-) Calculus and Fractional q-Calculus and Their Applications in Geometric Function Theory of Complex Analysis. Iranian Journal of Science and Technology, Transactions A: Science. 2020;44(1):327–344. Available from: https://doi.org/10.1007/s40995-019-00815-0
Wang B, Srivastava R, Liu JL. A Certain Subclass of Multivalent Analytic Functions Defined by the q-Difference Operator Related to the Janowski Functions. Mathematics. 2021;9(14):1706. Available from: https://doi.org/10.3390/math9141706
Altınkaya S, Yalçın S. CERTAIN CLASSES OF BI-UNIVALENT FUNCTIONS OF COMPLEX ORDER ASSOCIATED WITH QUASI-SUBORDINATION INVOLVING (p, q) -DERIVATIVE OPERATOR. Kragujevac Journal of Mathematics. 2020;44(4):639–649. Available from: https://doi./10.46793/KgJMat2004.639A
Araci S, Duran U, Acikgoz M, Srivastava HM. A certain ( p , q ) - derivative operator and associated divided differences. Journal of Inequalities and Applications. 2016;2016(1):301. Available from: https://doi:10.1186/s13660-016-1240-8
Srivastava HM, Raza N, Abujarad ESA, Srivastava GM, Abujarad MH. Fekete-Szegö inequality for classes of (p, q)-Starlike and (p, q)-convex functions. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas. 2019;113(4):3563–3584. Available from: https://doi.org/10.1007/s13398-019-00713-5
Al-Hawary T, Yousef F, Frasin BA. Subclasses of Analytic Functions of Complex Order Involving Jackson’s (p, q)-derivative. SSRN Electronic Journal. 2018. Available from: https:// doi.org/10.2139/ssrn.3289803
Acar T, Aral A, Mohiuddine SA. On Kantorovich modification of ( p , q ) -Baskakov operators. Journal of Inequalities and Applications. 2016;98(1):1–14. Available from: https://doi.org/10.1186/s13660-016-1045-9
Sanford S, Miller, Petru T, Mocanu. Differential subordinations: theory and applications. CRC Press. 2000.
Janteng A, Hern ALP, Omar R. Fekete-Szegö functional of classes of analytic functions involving the q-derivative operator. Applied Mathematical Sciences. 2020;14(10):481–488. Available from: https://doi.org/10.12988/ams.2020.914222
Fekete M, Szegö G. Eine Bemerkung Über Ungerade Schlichte Funktionen. Journal of the London Mathematical Society. 1933;1(2):85–89. Available from: https://doi.org/10.1112/jlms/s1-8.2.85
Ma W, Minda D. A unified treatment of some special classes of univalent functions. In: Proceedings of the Conference on Complex Analysis. (pp. 157-169) 1992.

Copyright

© 2022 Shilpa. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Published By Indian Society for Education and Environment (iSee)