The Fekete-Szegö Inequality and Successive Coefficients Difference for a Subclass of Close-to-Starlike Mappings in Complex Banach Spaces

Let C be the familiar class of normalized close-to-convex functions in the unit disk. In [17], Koepf demonstrated that, as to a function f ( ξ ) = ξ + ∑ m = 2 ∞ a m ξ m in the class C , max f ∈ C ∣ a 3 − λ a 2 2 ∣ ≤ { 3 − 4 λ , λ ∈ [ 0 , 1 3 ] , 1 3 + 4 9 λ , λ ∈ [ 1 3 , 2 3 ] , 1 , λ ∈ [ 2 3 , 1 ] . By applying this inequality, it can be proven that ∥ a 3 ∣−∣ a 2 ∥ ≤ 1 for close-to-convex functions. Now we generalized the above conclusions to a subclass of close-to-starlike mappings defined on the unit ball of a complex Banach space.