Geometric properties of harmonic functions associated with the normalized Mittag–Leffler function

Mohammad Faisal Khan; Muqrin A. Almuqrin; Mohammed AbaOud; Mohammad Faisal Khan; Muqrin A. Almuqrin; Mohammed AbaOud

doi:10.3934/math.20261278

AIMS Mathematics

2026, Volume 11, Issue 10: 32484-32513. doi: 10.3934/math.20261278

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Research article

Geometric properties of harmonic functions associated with the normalized Mittag–Leffler function

1.
Department of Basic Sciences, College of Science and Theoretical Studies, Prince Musaid bin Abdulrahman University, Riyadh 11673, Saudi Arabia; f.khan@seu.edu.sa
2.
Department of Mathematics, College of Science in Zulfi, Majmaah University, Al-Majmaah 11952, Saudi Arabia; m.almuqrin@mu.edu.sa
3.
Department of Mathematics and Statistics, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11564, Saudia Arabia; maabaoud@imamu.edu.sa

Received: 14 July 2026 Revised: 01 September 2026 Accepted: 08 September 2026 Published: 09 October 2026
MSC : Primary 30C45, 30C50, Secondary 05A30, 33E12

In this paper, we introduced and investigated a new subclass of complex-valued harmonic functions associated with the normalized two-parameter Mittag–Leffler function. The proposed class was defined by a second-order differential inequality involving the normalized Mittag–Leffler kernel, thereby establishing a direct connection between harmonic mapping theory and Mittag–Leffler-type special functions. An analytic characterization of the proposed harmonic class was derived through a family of analytic functions $ \Upsilon_{\varepsilon} = \eta+\varepsilon\xi $, where $ |\varepsilon| = 1 $. Furthermore, inclusion relations, parameter-dependent coefficient estimates, growth estimates, and closure under convex combinations were obtained. A Fekete–Szegő type inequality was also derived for the associated family of analytic functions. Several examples and graphical illustrations are included to demonstrate the geometric behavior of the proposed class. This paper unifies several known harmonic classes via differential inequalities and introduces a new framework for harmonic mappings with Mittag–Leffler and related functions.
- univalent functions,
- harmonic mappings,
- Mittag–Leffler function,
- differential inequality,
- starlike function,
- close-to-convex function,
- coefficient estimates,
- growth theorem,
- convex combinations,
- Fekete-Szegő inequality
Citation: Mohammad Faisal Khan, Muqrin A. Almuqrin, Mohammed AbaOud. Geometric properties of harmonic functions associated with the normalized Mittag–Leffler function[J]. AIMS Mathematics, 2026, 11(10): 32484-32513. doi: 10.3934/math.20261278

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Abstract

In this paper, we introduced and investigated a new subclass of complex-valued harmonic functions associated with the normalized two-parameter Mittag–Leffler function. The proposed class was defined by a second-order differential inequality involving the normalized Mittag–Leffler kernel, thereby establishing a direct connection between harmonic mapping theory and Mittag–Leffler-type special functions. An analytic characterization of the proposed harmonic class was derived through a family of analytic functions $ \Upsilon_{\varepsilon} = \eta+\varepsilon\xi $, where $ |\varepsilon| = 1 $. Furthermore, inclusion relations, parameter-dependent coefficient estimates, growth estimates, and closure under convex combinations were obtained. A Fekete–Szegő type inequality was also derived for the associated family of analytic functions. Several examples and graphical illustrations are included to demonstrate the geometric behavior of the proposed class. This paper unifies several known harmonic classes via differential inequalities and introduces a new framework for harmonic mappings with Mittag–Leffler and related functions.

References

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