Some refinements of the Fekete–Szegö inequalities for a class of biholomorphic mappings

Abstract

Let B be the unit disk U in C or the unit ball of a general complex Banach space. In this paper, we obtain a refinement of the Fekete–Szegö inequality for the class of S k + 1 0 ( B ) of mappings f for which there exist g-Loewner chains f ( z , t ) , where g ( ξ ) = 1 + ξ 1 − ξ , ξ ∈ U , such that f ( z ) = f ( z , 0 ) and z = 0 is a zero of order k + 1 of e − t f ( z , t ) − z for each t ≥ 0 . The results presented generalize the classical Fekete–Szegö inequality and the results in Hamada et al. (Fekete–Szegö problem for univalent mappings in one and higher dimensions. J Math Anal Appl. 2022;516:126526).