A comprehensive family of bi-univalent functions defined by (m, n)-Lucas polynomials

Making use of (m, n)-Lucas polynomials, we propose a comprehensive family of regular functions of the type g(z)=z+∑j=2∞djzj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g(z)=z+\sum \nolimits _{j=2}^{\infty }d_jz^j$$\end{document} which are bi-univalent in the disc {z∈C:|z|<1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{z\in {\mathbb {C}}:|z| <1\}$$\end{document}. We find estimates on the coefficients |d2|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|d_2|$$\end{document}, |d3|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|d_3|$$\end{document} and the of Fekete–Szegö functional for members of this subfamily. Relevant connections to existing results and new consequences of the main result are also presented.