A New Subclass of Univalent Functions Using Chebyshev Polynomials

Abstract

Background:

Geometric Function Theory (GFT) is a vital area in complex analysis, offering powerful techniques for solving physical and engineering problems. Particularly in two-dimensional problems, isomorphic mappings play a crucial role in simplifying complex regions, enabling more tractable analysis.

 Materials and Methods:

This study investigates key topics within GFT, focusing on a -generalized subclass of univalent functions . Chebyshev polynomials of the second kind are employed to estimate bounds for the coefficients of functions in this subclass. Theoretical tools from mathematical analysis and quantum calculus are integrated into the study to support this exploration.

Results:

We derive estimate coefficient bounds for functions in the  class and provide estimations for two important functionals: the Fekete–Szegö functional and the second-order Hankel determinant. These findings offer deeper insight into the geometric properties of the considered function class.

Conclusion:

The work strengthens the connection between geometric function theory and quantum calculus by introducing and analyzing a new subclass of univalent functions. The methods and estimations developed here enhance the understanding of analytic function behavior and open pathways for future work in applied mathematics and engineering models.