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

Received: 14 August 2023/Accepted: 2 January 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)

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*Corresponding author; email: mamoun@bau.edu.jo