This paper is concerned with the problem of estimating |a4 − a2a3| , where a k are the coefficients of a given close-to-convex function. The bounds of this expression for various classes of analytic functions have been applied to estimate the third Hankel determinant H3(1). The results for two subclasses of the class C of all close-to-convex functions are sharp. This bound is equal to 2. It is conjectured that this number is also the exact bound of |a4 − a2a3| for the whole class C .

In this paper, some coefficient problems for starlike analytic functions with respect to symmetric points are considered. Bounds of several coefficient functionals for functions belonging to this class are provided. The main aim of this paper is to find estimates for the following: coefficients, logarithmic coefficients, some cases of the generalized Zalcman coefficient functional, and some cases of the Hankel determinant.

We consider the order of strongly starlikeness in the class ofWe consider also the same problem for α-convex functions of order β.

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.

We consider the class S Ã ðf; aÞ, 0 a\1, of normalized analytic functions f such that Re zd f f ðzÞ f ðzÞ & ' [ a; jzj\1; where d f f is the convolution operator d f f ðzÞ ¼ 1 z f ðzÞ Ã z ð1 À fzÞð1 À zÞ & ' ; where f is complex, jfj 1. For f ¼ 1 the operator becomes the derivative f 0 , while for real f ¼ q, 0\q\1, we obtain the Jackson q-derivative d q f .