Bi-univalent functions of order \(\zeta\) connected with \((m,n)\)-Lucas polynomials

Volume 31, Issue 4, pp 433--447 http://dx.doi.org/10.22436/jmcs.031.04.06

Publication Date: May 24, 2023 Submission Date: March 20, 2023 Revision Date: April 04, 2023 Accteptance Date: May 10, 2023

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Authors

S. H. Hadi - Department of Mathematics, College of Education for Pure Sciences, University of Basrah, Basrah 61001, Iraq. - Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor Darul Ehsan, Malaysia. M. Darus - Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor Darul Ehsan, Malaysia. T. Bulboacă - Faculty of Mathematics and Computer Science, Babeș-Bolyai University, 400084 Cluj-Napoca, Romania.

Abstract

With the aid of the \(q\)-binomial coefficients and utilizing the convolution, we define a new \(q\)-convolution operator that helps us introduce two new families of bi-univalent functions. These classes are connected by subordination with a function \(\mathcal{G}_{m,n}\). We give upper bounds for the coefficients estimate \(|a_j| \ (j=2,3)\) of the functions that belong to these families, followed by some special cases. Moreover, we found estimates for the Fekete-Sezgö inequality for both of these families, followed by simple particular results. We emphasize that the defined convolution \(q\)-difference operator generalizes some other operators given by several authors. As an application of this study, Fekete-Sezgö inequalities for these classes of functions defined by Pascal distribution are investigated.

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Keywords

• Bi-univalent function
• \(q\)-exponential function
• \((m,n)\)-Lucas polynomials
• Fekete-Sezgö inequality
• Pascal distribution

MSC

•  30C45
•  33C45
•  11B39

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