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On the Fekete–Szegö Type Functionals for Close-to-Convex Functions

Mechanical Engineering Faculty, Lublin University of Technology, ul. Nadbystrzycka 36, 20-618 Lublin, Poland
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Symmetry 2019, 11(12), 1497; https://doi.org/10.3390/sym11121497
Received: 28 October 2019 / Revised: 29 November 2019 / Accepted: 7 December 2019 / Published: 10 December 2019
In this paper, we consider two functionals of the Fekete–Szegö type Θ f ( μ ) = a 4 μ a 2 a 3 and Φ f ( μ ) = a 2 a 4 μ a 3 2 for a real number μ and for an analytic function f ( z ) = z + a 2 z 2 + a 3 z 3 + , | z | < 1 . This type of research was initiated by Hayami and Owa in 2010. They obtained results for functions satisfying one of the conditions Re f ( z ) / z > α or Re f ( z ) > α , α [ 0 , 1 ) . Similar estimates were also derived for univalent starlike functions and for univalent convex functions. We discuss Θ f ( μ ) and Φ f ( μ ) for close-to-convex functions such that f ( z ) = h ( z ) / ( 1 z ) 2 , where h is an analytic function with a positive real part. Many coefficient problems, among others estimating of Θ f ( μ ) , Φ f ( μ ) or the Hankel determinants for close-to-convex functions or univalent functions, are not solved yet. Our results broaden the scope of theoretical results connected with these functionals defined for different subclasses of analytic univalent functions. View Full-Text
Keywords: coefficient problem; close-to-convex function; Fekete–Szegö functional; functional of Fekete–Szegö type coefficient problem; close-to-convex function; Fekete–Szegö functional; functional of Fekete–Szegö type
MDPI and ACS Style

Tra̧bka-Wiȩcław, K.; Zaprawa, P.; Gregorczyk, M.; Rysak, A. On the Fekete–Szegö Type Functionals for Close-to-Convex Functions. Symmetry 2019, 11, 1497.

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