On Geometric Properties of k-Mittag–Leffler Function
- 1Department of Mathematics, Riphah International University, Islamabad 44000, Pakistan
- 2Faculty of Exact and Technical Sciences, University of Rzeszów, 35-310 Rzeszów, Poland
Abstract
This paper investigates geometric properties of a normalized k-Mittag–Leffler function, where k>0 is a fixed real parameter, involving the k-Gamma function and the k-Pochhammer symbol. Since the coefficient structure of this function differs essentially from that of the classical Mittag–Leffler and Le Roy-type Mittag–Leffler functions, the standard monotonicity arguments available in the literature cannot be applied directly. To overcome this difficulty, we first derive a two-sided inequality for the k-digamma function and use it to establish suitable monotonicity properties of the associated coefficient sequences. These auxiliary results form the main analytical foundation of the paper. By combining them with classical criteria from geometric function theory, we obtain explicit sufficient conditions for the normalized k-Mittag–Leffler function to be starlike, parabolic starlike, convex, uniformly convex, and close-to-convex in the unit disk. Corresponding starlikeness and convexity results in the half-unit disk are also established. In addition, we study an associated odd normalized function and derive conditions ensuring its close-to-convexity with respect to 12log1+τ1−τ. Several corollaries and examples are presented to analyze the applicability of the results. For k=1, the obtained assertions reduce to the corresponding results for the normalized Prabhakar (three-parameter) Mittag–Leffler function, and for k=q=1 to those for the normalized two-parameter Mittag–Leffler function.
Related
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- A Nonlinear Fractional Quantum Coupled System: Existence and Mittag-Leffler Ulam-Hyers Stability
- Novel Fractional Delayed Mittag-Leffler-Type Matrices and Their Applications Without Commutativity Constraints
- A sharp Mittag-Leffler bound for series with separated exponents
- Fractional q-Starlikeness in Mittag–Leffler and Limaçon Domains
BibTeX
@article{khan2026geometric,
title = {{On Geometric Properties of k-Mittag–Leffler Function}},
author = {Shahid Khan and Janusz Sokół},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14213968},
url = {https://www.mdpi.com/2227-7390/14/21/3968},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - On Geometric Properties of k-Mittag–Leffler Function
AU - Shahid Khan
AU - Janusz Sokół
PY - 2026
DA - 2026/10/30
JO - Mathematics
DO - 10.3390/math14213968
UR - https://www.mdpi.com/2227-7390/14/21/3968
AB - This paper investigates geometric properties of a normalized k-Mittag–Leffler function, where k>0 is a fixed real parameter, involving the k-Gamma function and the k-Pochhammer symbol. Since the coefficient structure of this function differs essentially from that of the classical Mittag–Leffler and Le Roy-type Mittag–Leffler functions, the standard monotonicity arguments available in the literature cannot be applied directly. To overcome this difficulty, we first derive a two-sided inequality for the k-digamma function and use it to establish suitable monotonicity properties of the associated coefficient sequences. These auxiliary results form the main analytical foundation of the paper. By combining them with classical criteria from geometric function theory, we obtain explicit sufficient conditions for the normalized k-Mittag–Leffler function to be starlike, parabolic starlike, convex, uniformly convex, and close-to-convex in the unit disk. Corresponding starlikeness and convexity results in the half-unit disk are also established. In addition, we study an associated odd normalized function and derive conditions ensuring its close-to-convexity with respect to 12log1+τ1−τ. Several corollaries and examples are presented to analyze the applicability of the results. For k=1, the obtained assertions reduce to the corresponding results for the normalized Prabhakar (three-parameter) Mittag–Leffler function, and for k=q=1 to those for the normalized two-parameter Mittag–Leffler function.
SN - 2227-7390
ER -