Semnan University International Journal of Nonlinear Analysis and Applications 2008-6822 2025 11 20 Comprehensive subclass of univalent functions associated with the Gegenbauer polynomial and the Bahalola operator 10263 10.22075/ijnaa.2025.35692.5307 EN Sahar Ahmadpour Department of Mathematics, Payame Noor University, P. O. Box: 19395-3697, Tehran, Iran Shahram Najafzadeh Department of Mathematics, Payame Noor University, P.O. Box: 19395-3697, Tehran, Iran Mohammadreza Foroutan Department of Mathematics, Payame Noor University, P. O. Box: 19395-3697, Tehran, Iran Robab Haghi Department of Mathematics, Payame Noor University, P. O. Box: 19395-3697, Tehran, Iran Journal Article 2024 10 22 Our objective in this paper is to introduce a newly constructed subclass of univalent functions. The Gegenbauer Polynomials are very successful in the theory of polynomials because they extend many polynomials, such as Legendre, Chebyshev polynomials, etc. Also, the Babolala operator for a fixed real number is used. Furthermore, by applying the convolution structures and subordination property, the new subclass is obtaind. For this new subclass, first, the share coefficient bounds are determined. These coefficient estimates are very important to obtain the geometric properties. So after that, we obtain the radii of starlikeness, convexity and close-to-convexity. Also, under some restrictions on parameters, we show that the newly defined subclass has a convolution-preserving property. Finally, the convexity of this subclass is pointed out. Our objective in this paper is to introduce a newly constructed subclass of univalent functions. The Gegenbauer Polynomials are very successful in the theory of polynomials because they extend many polynomials, such as Legendre, Chebyshev polynomials, etc. Also, the Babolala operator for a fixed real number is used. Furthermore, by applying the convolution structures and subordination property, the new subclass is obtaind. For this new subclass, first, the share coefficient bounds are determined. These coefficient estimates are very important to obtain the geometric properties. So after that, we obtain the radii of starlikeness, convexity and close-to-convexity. Also, under some restrictions on parameters, we show that the newly defined subclass has a convolution-preserving property. Finally, the convexity of this subclass is pointed out. https://ijnaa.semnan.ac.ir/article_10263_846ee3b1e945a9f545d13db93bb8fa6c.pdf