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