Fekete and Szegö inequality for a subclass of almost spirallike mappings of type β and order α on the bounded starlike circular domain in ℂ n

Abstract

Let S ^ ( α , β ) be the familiar class of almost spirallike functions of type β and order α in the unit disk (see Definition 1.1). In this paper, first, we prove that for a function f ( z ) = z + ∑ n = 2 ∞ a n z n in the class S ^ ( α , β ) , then | a 3 − λ a 2 2 | ≤ ( 1 − α ) cos ⁡ β max { 1 , | 1 − 4 ( 1 − α ) ( 1 − λ ) cos ⁡ β e i β | } , λ ∈ C . The above estimation is sharp. Second, we extend this result to the bounded starlike circular domain in C n and obtain the sharp estimates. The results presented here would provide extensions of those given by Xu et al. [The Fekete and Szegö problem on the bounded starlike circular domain in C n . Pure Appl Math Q. 2016;12:621–638] and Xiong [Sharp coefficients bounds for class of almost starlike mappings of order α in C n . J Math Inequalities. 2020;14:853–865]. Finally, a certain conjecture is also formulated.