On $ q $-differential operators acting on bounded turning functions associated with epicycloidal curves

Isra Al-Shbeil; Wael Mahmoud Mohammad Salameh; Maha Alammari; Alina Alb Lupas; Isra Al-Shbeil; Wael Mahmoud Mohammad Salameh; Maha Alammari; Alina Alb Lupas

doi:10.3934/math.2026803

AIMS Mathematics

2026, Volume 11, Issue 7: 19784-19807. doi: 10.3934/math.2026803

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On $ q $-differential operators acting on bounded turning functions associated with epicycloidal curves

1.
Department of Mathematics, Faculty of Science, University of Jordan, Amman 11942, Jordan; i.shbeil@ju.edu.jo
2.
Faculty of Information Technology, Abu Dhabi University, Abu Dhabi 59911, United Arab Emirates; wael.salameh@adu.ac.ae
3.
Department of Mathematics, College of Science, King Saud University, P.O. Box 22452 Riyadh 11495, Saudi Arabia; malammari@ksu.edu.sa
4.
Department of Mathematics and Computer Science, University of Oradea, 410087 Oradea, Romania; alb@uoradea.ro

Received: 05 May 2026 Revised: 11 June 2026 Accepted: 23 June 2026 Published: 07 July 2026
MSC : 30C50, 30C80, 30C45, 47B38, 11B65

The $ q $-calculus has found numerous applications in engineering, particularly in mass and heat transfer problems, as well as in the study of nonlinear and fuzzy differential equations arising in science and technology. The $ q $-difference operator $ \mathfrak{D}_q $ and related $ q $-calculus operators play an important role in the development of this theory. In this paper, we introduce and investigate a new class of analytic functions associated with epicycloidal domains by means of subordination theory and a $ q $-differential operator. Our primary objective is to characterize this class and establish several of its geometric properties in the unit disk $ \mathbb{U} $. In particular, we study coefficient estimates, the Zalcman conjecture and its generalized form, the second and third Hankel determinants, the Krushkal inequality, and the Fekete–Szegö functional.
- $ q $-differential operator,
- Fekete–Szegöproblem,
- Zalcman conjecture,
- Hankel determinant,
- Krushkal inequality,
- $ q $-derivative,
- bounded turning function,
- epicycloidal domain,
- quantum calculus,
- analytic functions
Citation: Isra Al-Shbeil, Wael Mahmoud Mohammad Salameh, Maha Alammari, Alina Alb Lupas. On $ q $-differential operators acting on bounded turning functions associated with epicycloidal curves[J]. AIMS Mathematics, 2026, 11(7): 19784-19807. doi: 10.3934/math.2026803

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Abstract

The $ q $-calculus has found numerous applications in engineering, particularly in mass and heat transfer problems, as well as in the study of nonlinear and fuzzy differential equations arising in science and technology. The $ q $-difference operator $ \mathfrak{D}_q $ and related $ q $-calculus operators play an important role in the development of this theory. In this paper, we introduce and investigate a new class of analytic functions associated with epicycloidal domains by means of subordination theory and a $ q $-differential operator. Our primary objective is to characterize this class and establish several of its geometric properties in the unit disk $ \mathbb{U} $. In particular, we study coefficient estimates, the Zalcman conjecture and its generalized form, the second and third Hankel determinants, the Krushkal inequality, and the Fekete–Szegö functional.

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