## Bounds on Initial Coefficients for a Certain New Subclass of Bi-univalent Functions by Means of Faber Polynomial Expansions

Mathematics in Computer Science, vol.13, no.3, pp.441-447, 2019 (Refereed Journals of Other Institutions)

• Publication Type: Article / Article
• Volume: 13 Issue: 3
• Publication Date: 2019
• Doi Number: 10.1007/s11786-019-00406-7
• Title of Journal : Mathematics in Computer Science
• Page Numbers: pp.441-447

#### Abstract

In this paper, we present a new subclass ${\mathcal{T}}_{\Sigma }\left(\mu \right)$ of bi univalent functions belong to $\Sigma$ in the open unit disc $\mathcal{U}=\left\{z\phantom{\rule{thinmathspace}{0ex}}:\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}z\in \mathcal{C}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}|z|<1\right\}$. Then, we use the concepts of Faber polynomial expansions to find upper bound for the general coefficient of such functions belongs to the defined class. Further, for the functions in this subclass we obtain bound on first three coefficients $|{a}_{2}|$$|{a}_{3}|$ and $|{a}_{4}|$. We hope that this paper will inspire future researchers in applying our approach to other related problems.

In this paper, we present a new subclass ${\mathcal{T}}_{\Sigma }\left(\mu \right)$ of bi univalent functions belong to $\Sigma$ in the open unit disc $\mathcal{U}=\left\{z\phantom{\rule{thinmathspace}{0ex}}:\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}z\in \mathcal{C}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}|z|<1\right\}$. Then, we use the concepts of Faber polynomial expansions to find upper bound for the general coefficient of such functions belongs to the defined class. Further, for the functions in this subclass we obtain bound on first three coefficients $|{a}_{2}|$$|{a}_{3}|$ and $|{a}_{4}|$. We hope that this paper will inspire future researchers in applying our approach to other related problems.