Next Article in Journal
Combined Conflict Evidence Based on Two-Tuple IOWA Operators
Previous Article in Journal
Statistical Criteria of Nanofluid Flows over a Stretching Sheet with the Effects of Magnetic Field and Viscous Dissipation
Previous Article in Special Issue
A Note on the Degenerate Type of Complex Appell Polynomials
Open AccessArticle

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

by Oh Sang Kwon 1, Shahid Khan 2 , Young Jae Sim 1,* and Saqib Hussain 3
1
Department of Mathematics, Kyungsung University, Busan 48434, Korea
2
Department of Mathematics, Riphah International University, Islamabad 44000, Pakistan
3
Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad 22060, Pakistan
*
Author to whom correspondence should be addressed.
Symmetry 2019, 11(11), 1368; https://doi.org/10.3390/sym11111368
Received: 30 August 2019 / Revised: 11 October 2019 / Accepted: 16 October 2019 / Published: 4 November 2019
Let $Σ$ be the class of meromorphic functions f of the form $f ( ζ ) = ζ + ∑ n = 0 ∞ a n ζ − n$ which are analytic in $Δ : = { ζ ∈ C : | ζ | > 1 }$ . For $n ∈ N 0 : = N ∪ { 0 }$ , the nth Faber polynomial $Φ n ( w )$ of $f ∈ Σ$ is a monic polynomial of degree n that is generated by a function $ζ f ′ ( ζ ) / ( f ( ζ ) − w )$ . For given $f ∈ Σ$ , by $F n , i ( f )$ , we denote the ith coefficient of $Φ n ( w )$ . For given $0 ≤ α < 1$ and $0 < β ≤ 1$ , let us consider domains $H α$ and $S β ⊂ C$ defined by $H α = { w ∈ C : Re ( w ) > α }$ and $S β = { w ∈ C : | arg ( w ) | < β }$ , which are symmetric with respect to the real axis. A function $f ∈ Σ$ is called meromorphic starlike of order $α$ if $ζ f ′ ( ζ ) / f ( ζ ) ∈ H α$ for all $ζ ∈ Δ$ . Another function $f ∈ Σ$ is called meromorphic strongly starlike of order $β$ if $ζ f ′ ( ζ ) / f ( ζ ) ∈ S β$ for all $ζ ∈ Δ$ . In this paper we investigate the sharp bounds of $F n , n − i ( f )$ , $n ∈ N 0$ , $i ∈ { 2 , 3 , 4 }$ , for meromorphic starlike functions of order $α$ and meromorphic strongly starlike of order $β$ . Similar estimates for meromorphic convex functions of order $α$ ( $0 ≤ α < 1$ ) and meromorphic strongly convex of order $β$ ( $0 < β ≤ 1$ ) are also discussed. View Full-Text
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.