Sains Malaysiana 53(1)(2024): 231-238

http://doi.org/10.17576/jsm-2024-5301-18

Properties for Certain Class of -p Valent Functions Related to Jackson’s Operator

(Sifat bagi Kelas Fungsi Tertentu Valen -p yang Berkaitan dengan Pengoperasi Jackson)

MA’MOUN I.Y. ALHARAYZEH^1,*, MASLINA DARUS², FAISAL Y. ALZYOUD³ & HABIS S. AL-ZBOON⁴

¹Department of Scientific Basic Sciences, Faculty of Engineering Technology, Al-Balqa Applied University, Amman 11134, Jordan

 ²Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia

 ³Department of Computer Science, Faculty of Information Technology, Isra University, Amman 1162, Jordan

⁴Department of Education and Cultural Studies, College of Arts and Sciences, University of Nizwa, Oman

Diserahkan: 14 Ogos 2023/Diterima: 2 Januari 2024

Abstract

An inspiration from the fundamentals of (r,q) calculus to introduce an innovative subclass within the T(p) category of multivalent analytic functions, located within the confines of the open unit disk, is subjected to examination. The establishment of the subclass was achieved by employing Jackson's derivative operator to enhance the comprehension of these analytical functions. This article began by investigating and establishing adequate criteria that dictate the inclusion of functions within this recently introduced subclass. To achieve this, a comprehensive coefficient characterization to facilitate a deeper comprehension of the subclass's properties and behavior is derived. Further, various pertinent results that contribute to the broader understanding of the functions belonging to this subclass are explored. The findings and implications of these results are elucidated, underscoring the potential significance of this work in advancing the field of multivalent analytic functions and their applications. In conclusion, this paper broadens the scope of T(p) and sheds light on the distinct characteristics exhibited by the functions in this newly introduced subclass. This work sets the stage for further exploration and applications of (r,q) calculus and Jackson's derivative operator in the domain of multivalent analytic functions.

Keywords: p- valent function; quantum or (r,q) -calculus; (r,q) -derivative operator

Abstrak

Inspirasi daripada konsep asas kalkulus- (r,q) dengan memperkenalkan satu subkelas yang inovatif dalam kategori fungsi analisis multivalen T(p) tertakrif pada cakera unit terbuka akan dikaji. Pembinaan subkelas tercapai dengan menggunakan pengoperasi terbitan Jackson bagi meningkatkan kefahaman berkenaan fungsi analisis ini. Makalah ini dimulakan dengan mengkaji dan membina kriteria yang cukup bagi menyatakan rangkuman fungsi agar terkandung dalam subkelas yang diperkenalkan. Bagi mencapai hasrat tersebut, satu ciri pekali yang komprehensif diperoleh bagi memudahkan lagi kefahaman terhadap sifat dan kelakuan subkelas tersebut. Malah beberapa hasil penting yang menyumbang kepada kefahaman luas bagi subkelas fungsi ini dikaji. Keputusan dan implikasi hasil ini diperjelaskan dan menekankan potensi kepentingan kajian dalam memajukan bidang fungsi analisis multivalen dan penggunaannya. Kesimpulannya, makalah ini memperluaskan skop T(p) dan memberi pencerahan kepada ciri berbeza yang ditunjukkan oleh fungsi dalam subkelas baharu yang diperkenalkan. Kajian ini menyediakan ruang kepada penerokaan lanjutan dan penggunaan kalkulus- (r,q) dan pengoperasi terbitan Jackson dalam domain fungsi analisis multivalen.

Kata kunci:  Fungsi valen - p ; kuantum atau kalkulus- (r,q) ; pengoperasi terbitan -(r,q)

RUJUKAN

Alharayzeh, M.I. 2021. On a subclass of k-uniformly analytic and multivalent functions defined by q-calculus operator. Far East Journal of Mathematical Sciences 132(1): 1-20.

Alharayzeh, M.I. & Alzboon, H.S. 2023. On a subclass of k-uniformly analytic functions with negative coefficients and their properties. Nonlinear Functional Analysis and Applications 28(2): 589-599.

Alharayzeh, M.I. & Ghanim, F. 2022. New subclass of k-uniformly univalent analytic functions with negative coefficients defined by multiplier transformation. Abstract and Applied Analysis 2022: Article ID. 4593799.

Alharayzeh, M. & Darus, M. 2010. On subclass of analytic univalent functions associated with negative coefficients. International Journal of Mathematics and Mathematical 2010: Article ID. 343580.

Aqlan, E., Jahangiri, J.M. & Kulkarni, S.R. 2004. New classes of $ k $-uniformly convex and starlike functions. Tamkang Journal of Mathematics 35(3): 261-266.

Chakrabarti, R. & Jagannathan, R. 1991. A (p, q)-oscillator realization of two-parameter quantum algebras. Journal of Physics A: Mathematical and General 24(13): L711-L718.

Dixit, K.K. & Pal, S.K. 1995. On a class of univalent functions related to complex order. Indian Journal of Pure and Applied Mathematics 26(9): 889-896.

Harayzeh, M. & Darus, M. 2011. The Fekete-Szegö theorem for a certain class of analytic functions. Sains Malaysiana 40(4): 385-389.

Ismail, M.E.H., Merkes, E. & Steyr, D. 1990. A generalization of starlike functions. Complex Variables, Theory and Application: An International Journal 14(1-4): 77-84.

Jackson, F.H. 1910. On a q-definite integrals. The Quarterly Journal of Pure and Applied Mathematics 41: 193-203.‏

Jackson, F.H. 1909. On q-functions and a certain difference operator. Earth and Environmental Science Transactions of the Royal Society of Edinburgh 46(2): 253-281.

Kanas, S. & Răducanu, D. 2014. Some subclass of analytic functions related to conic domains. Mathematica Slovaca 64(5): 1183-1196.‏

Kunt, M., Iscan, I., Alp, N. & Sarikaya, M. 2018. (p, q)-Hermite-Hadamard inequalities and (p, q)- estimates for midpoint type inequalities via convex and quasi-convex functions. Revista De La Real Academia De Ciencias Exactas Fisicas Y Naturales Serie A-Matematicas 112(4): 969-992.

Prabseang, J., Nonlaopon, K. & Tariboon, J. 2019. (p, q)-Hermite–Hadamard inequalities for double integral and (p, q)-differentiable convex functions. Axioms 8(2): 68.‏

Sadjang, P.N. 2018. On the fundamental theorem of (p, q)-calculus and some (p, q)-Taylor formulas. Results in Mathematics 73: 39.‏

Silverman, H. 1975. Univalent functions with negative coefficients. Proceedings of the American Mathematical Society 51(1): 109-116.‏

Tunç, M. & Göv, E. 2021. Some integral inequalities via (p, q)-calculus on finite intervals. Filomat 35(5): 1421-1430.

*Pengarang untuk surat-menyurat; email: mamoun@bau.edu.jo