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Peer-reviewedCC BY 4.0Fractal and Fractional

Fractional q-Starlikeness in Mittag–Leffler and Limaçon Domains

  • Hussain Gissy1
  • Mohamed Biomy2
  • Mirajul Haq3
  • Muhammad Arif3
  • Muhammad Abbas3
  1. 1Department of Mathematics, Faculty of Science, Jazan University, Jazan 45142, Saudi Arabia
  2. 2Department of Management Information Systems, College of Business and Economics, Qassim University, Buraydah 51452, Saudi Arabia
  3. 3Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan
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Abstract

In this paper, we use a normalized fractional q-derivative operator, constructed by combining Jackson’s q-difference operator with a fractional q-integral, to introduce two subclasses of normalized analytic functions in the open unit disk whose fractional transforms are starlike and univalent. The first subclass, Sq,λ*(ML), is defined by subordination to a normalized q-Mittag–Leffler function, whereas the second, Sq,λ*(L), is associated with a generalized limaçon domain. For both classes, we investigate sharp initial coefficient bounds and sharp estimates for the Fekete–Szegö functional. We also study inclusion and radius relations between the two classes. This comparative approach clarifies how an analytically defined Mittag–Leffler domain and a geometrically defined limaçon domain affect coefficient estimates when the underlying operator is held fixed. Relevant classical and quantum calculus results are recovered through suitable parameter choices and in the limiting case q→1−.

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Cite

BibTeXRIS
BibTeX
@article{gissy2026fractional,
  title = {{Fractional q-Starlikeness in Mittag–Leffler and Limaçon Domains}},
  author = {Hussain Gissy and Mohamed Biomy and Mirajul Haq and Muhammad Arif and Muhammad Abbas},
  journal = {Fractal and Fractional},
  year = {2026},
  month = oct,
  doi = {10.3390/fractalfract10100720},
  url = {https://www.mdpi.com/2504-3110/10/10/720},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Fractional q-Starlikeness in Mittag–Leffler and Limaçon Domains
AU  - Hussain Gissy
AU  - Mohamed Biomy
AU  - Mirajul Haq
AU  - Muhammad Arif
AU  - Muhammad Abbas
PY  - 2026
DA  - 2026/10/14
JO  - Fractal and Fractional
DO  - 10.3390/fractalfract10100720
UR  - https://www.mdpi.com/2504-3110/10/10/720
AB  - In this paper, we use a normalized fractional q-derivative operator, constructed by combining Jackson’s q-difference operator with a fractional q-integral, to introduce two subclasses of normalized analytic functions in the open unit disk whose fractional transforms are starlike and univalent. The first subclass, Sq,λ*(ML), is defined by subordination to a normalized q-Mittag–Leffler function, whereas the second, Sq,λ*(L), is associated with a generalized limaçon domain. For both classes, we investigate sharp initial coefficient bounds and sharp estimates for the Fekete–Szegö functional. We also study inclusion and radius relations between the two classes. This comparative approach clarifies how an analytically defined Mittag–Leffler domain and a geometrically defined limaçon domain affect coefficient estimates when the underlying operator is held fixed. Relevant classical and quantum calculus results are recovered through suitable parameter choices and in the limiting case q→1−.
SN  - 2504-3110
ER  -