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

Abstract

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.