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

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 Sk+10(B) of mappings f for which there exist g-Loewner chains f(z,t), where g(ξ)=1+ξ1−ξ,ξ∈U, such that f(z)=f(z,0) and z = 0 is a zero of order k + 1 of e−tf(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).