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

Abstract

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.

Estimates of Coefficient Functionals for Functions Convex in the Imaginary-Axis Direction

“…The class of all functions satisfying (3) is denoted by C β (g) (see [9]). Coefficient problems for the class C 0 (k), where k is the Koebe function k(z) = z/(1 − z) 2 , were discussed in a few papers (see, for example, [10][11][12][13]).…”

On the Fekete–Szegö type functionals for functions which are convex in the direction of the imaginary axis

“…The functionals Φ f and Θ f for f in S * and in the class K of convex functions were discussed in [30]. The estimates of Φ f and Θ f for functions in C 0 (k) were published in [28].…”