Bounded support points for mappings with g-parametric representation in C2

Support points for families of univalent mappings on bounded symmetric domains

“…In this paper, we give positive answers to the above questions in the case the domain D is a bounded symmetric domain realized as the unit ball of an n-dimensional JB * -triple X = (C n , ∥ • ∥ X ) of rank r 2. For the proof, we use the convexity of g and also the property that there exists an r-dimensional subspace X 1 of C n such that B X ∩ X 1 may be regarded as the unit polydisc U r of dimension r. The main results complement recent extremal results for mappings with parametric representation on the unit ball B n (see [2,5,17,18,22]), and on the unit polydisc U n in C n (see [14,38]). Since we do not assume that a 0 (g) = dist(1, ∂g(U)), our result is an improvement of the above theorems.…”

“…These mappings may be embedded as the first elements of g-Loewner chains (see Definition 2.10). Sharp growth and coefficient bounds for the family S 0 g (B n ) were obtained in [12] and [18]. In particular, these results provide also sharp growth and coefficient estimates for various families of normalized univalent mappings on B n , such as starlike and convex mappings (see also [19] and the references therein).…”

“…and then the result contained in [18,Theorem 4.11 and Remark 4.14] is as follows. Note that if g satisfies the above conditions, then we have (see, e.g., [23,27]) min{ℜg(ζ) : |ζ| = ρ} = g(ρ), max{ℜg(ζ) : |ζ| = ρ} = g(−ρ), ∀ ρ ∈ (0, 1).…”

“…We begin this section with the following notion, which is an analogue on B X of the shearing process due to Bracci [2, Definition 1.3]. Then we obtain some coefficient bounds for the family M g (B X ), where g : U → C satisfies the conditions of Assumption 1.1 (see [2, Proposition 2.1], [14,Proposition 4.2] and [18,Proposition 4.2]). We also obtain various particular cases.…”

“…where the branch of the power function is chosen such that ( 1−ζ 1+ζ ) γ | ζ=0 = 1. Graham et al [18] generalized Bracci's result [2] to the case of mappings in S 0 g (B 2 ), and proved the existence of a bounded g-starlike mapping in the family S 0 g (B 2 ) of mappings with g-parametric representation on the Euclidean unit ball B 2 in C 2 which is a support point for a linear functional. In the case g satisfies the conditions of Assumption 1.1, g(ζ) = g(ζ) for all ζ ∈ U, g ′ (0) < 0 and a 0 (g) = dist(1, ∂g(U)), where a 0 (g) is given by a 0 (g) = inf…”