Published in Mathematics Journal, April 11, 2025
A new study by an international team of mathematicians, recently published in the journal Mathematics, presents groundbreaking insights into the structure of bi-univalent functions using the generalized (p,q)-derivative operator. The research marks a significant contribution to the field of complex analysis and could pave the way for further theoretical advancements in analytic function theory.
The collaborative work, titled “On the Third Hankel Determinant of a Certain Subclass of Bi-Univalent Functions Defined by (p,q)-Derivative Operator”, was authored by Mohammad El-Ityan (Jordan), Qasim Ali Shakir (Iraq), Tariq Al-Hawary (Jordan), Rafid Buti (Iraq), Daniel Breaz (Romania), and Luminita-Ioana Cotîrlă (Romania), who all contributed equally to the study.
Advancing the Theory of Bi-Univalent Functions
The paper introduces a new subclass of bi-univalent functions—functions that are univalent (one-to-one) in both the unit disk and its inverse domain—through the lens of the (p,q)-derivative operator, a generalization that adds a new layer of flexibility and depth to classical derivative methods.
By employing an operational method tailored to this context, the authors successfully established sharp coefficient estimates up to the fifth term (|ℓ₅|), shedding light on the geometric behavior and analytic structure of the newly defined function class. These estimates are critical in understanding the functional behavior within the unit disk, a central concern in geometric function theory.
Key Results: Hankel Determinants and Functional Estimates
The study particularly emphasizes the computation of Hankel determinants—specifically the second and third orders—which play a pivotal role in analyzing the growth, distortion, and coefficient bounds of analytic functions. The paper delivers explicit upper bounds for these determinants, along with results for the well-known Fekete–Szegö functional, a cornerstone in the coefficient theory of univalent functions.
These findings offer new tools for mathematicians working with special classes of analytic and bi-univalent functions and open new avenues for applying the (p,q)-calculus framework in complex analysis.
A Step Toward Broader Mathematical Exploration
According to the authors, this study not only deepens current understanding but also lays the foundation for further explorations. Researchers may use similar techniques to investigate other subclasses, potentially involving more generalized differential or integral operators. The paper is part of a broader special issue focusing on “Advanced Research in Complex Analysis Operators and Special Classes of Analytic Functions.”
The research team’s diverse institutional backgrounds—from Jordan, Iraq, and Romania—highlight the global collaboration driving innovation in pure mathematics.
For more details, the full article is available at: https://doi.org/10.3390/math13081269
Mathematics 2025, 13(8), 1269
