# Bounds for the Coefficient of Faber Polynomial of Meromorphic Starlike and Convex Functions

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Department of Mathematics, Kyungsung University, Busan 48434, Korea

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Department of Mathematics, Riphah International University, Islamabad 44000, Pakistan

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Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad 22060, Pakistan

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Author to whom correspondence should be addressed.

Received: 30 August 2019 / Revised: 11 October 2019 / Accepted: 16 October 2019 / Published: 4 November 2019

(This article belongs to the Special Issue The 32th Congress of The Jangjeon Mathematical Society (ICJMS2019) will be Held at Far Eastern Federal Universit, Vladivostok Russia)

Let $\mathsf{\Sigma}$ be the class of meromorphic functions f of the form $f\left(\zeta \right)=\zeta +{\sum}_{n=0}^{\infty}{a}_{n}{\zeta}^{-n}$ which are analytic in $\Delta :=\{\zeta \in \mathbb{C}:|\zeta |>1\}$ . For $n\in {\mathbb{N}}_{0}:=\mathbb{N}\cup \left\{0\right\}$ , the nth Faber polynomial ${\mathsf{\Phi}}_{n}\left(w\right)$ of $f\in \mathsf{\Sigma}$ is a monic polynomial of degree n that is generated by a function $\zeta {f}^{\prime}\left(\zeta \right)/(f\left(\zeta \right)-w)$ . For given $f\in \mathsf{\Sigma}$ , by ${F}_{n,i}\left(f\right)$ , we denote the ith coefficient of ${\mathsf{\Phi}}_{n}\left(w\right)$ . For given $0\le \alpha <1$ and $0<\beta \le 1$ , let us consider domains ${\mathbb{H}}_{\alpha}$ and ${S}_{\beta}\subset \mathbb{C}$ defined by ${\mathbb{H}}_{\alpha}=\{w\in \mathbb{C}:\mathrm{Re}\left(w\right)>\alpha \}$ and ${S}_{\beta}=\{w\in \mathbb{C}:|arg\left(w\right)|<\beta \}$ , which are symmetric with respect to the real axis. A function $f\in \mathsf{\Sigma}$ is called meromorphic starlike of order $\alpha $ if $\zeta {f}^{\prime}\left(\zeta \right)/f\left(\zeta \right)\in {\mathbb{H}}_{\alpha}$ for all $\zeta \in \Delta $ . Another function $f\in \mathsf{\Sigma}$ is called meromorphic strongly starlike of order $\beta $ if $\zeta {f}^{\prime}\left(\zeta \right)/f\left(\zeta \right)\in {S}_{\beta}$ for all $\zeta \in \Delta $ . In this paper we investigate the sharp bounds of ${F}_{n,n-i}\left(f\right)$ , $n\in {\mathbb{N}}_{0}$ , $i\in \{2,3,4\}$ , for meromorphic starlike functions of order $\alpha $ and meromorphic strongly starlike of order $\beta $ . Similar estimates for meromorphic convex functions of order $\alpha $ ( $0\le \alpha <1$ ) and meromorphic strongly convex of order $\beta $ ( $0<\beta \le 1$ ) are also discussed.
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*Keywords:*meromorphic functions; starlike functions; convex functions; Faber polynomials; coefficient problems

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**MDPI and ACS Style**

Kwon, O.S.; Khan, S.; Sim, Y.J.; Hussain, S. Bounds for the Coefficient of Faber Polynomial of Meromorphic Starlike and Convex Functions. *Symmetry* **2019**, *11*, 1368.

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