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Open AccessArticle

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
*
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
Keywords: meromorphic functions; starlike functions; convex functions; Faber polynomials; coefficient problems meromorphic functions; starlike functions; convex functions; Faber polynomials; coefficient problems
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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