Extremal Problems and g-Loewner Chains in \(\mathbb{C}^{n}\) and Reflexive Complex Banach Spaces

Let X be a reflexive complex Banach space with the unit ball B. In the first part of the paper, we survey various growth and coefficient bounds for mappings in the Carathéodory family \(\mathcal{M}\), which plays a key role in the study of the generalized Loewner differential equation. Then we consider recent results in the theory of Loewner chains and the generalized Loewner differential equation on the unit ball of \(\mathbb{C}^{n}\) and reflexive complex Banach spaces. In the second part of this paper, we obtain sharp growth theorems and second coefficient bounds for mappings with g-parametric representation and we present certain particular cases of special interest. Finally, we consider extremal problems related to bounded mappings in \(S_{g}^{0}(B^{n})\), where B ⁿ is the Euclidean unit ball in \(\mathbb{C}^{n}\). To this end, we use ideas from control theory to investigate the (normalized) time-logM-reachable family \(\tilde{\mathcal{R}}_{\log M}(\mathrm{id}_{B^{n}},\mathcal{M}_{g})\) generated by a subset \(\mathcal{M}_{g}\) of \(\mathcal{M}\), where M ≥ 1 and g is a univalent function on the unit disc U such that g(0) = 1, \(\mathfrak{R}g(\zeta ) > 0\), | ζ |  < 1, and which satisfies some natural conditions. We characterize this family in terms of univalent subordination chains, and we obtain certain results related to extreme points and support points associated with the compact family \(\overline{\tilde{\mathcal{R}}_{\log M}(\mathrm{id}_{B^{n}},\mathcal{M}_{g})}\). Also, we give some examples of mappings in \(\tilde{\mathcal{R}}_{\log M}(\mathrm{id}_{B^{n}},\mathcal{M}_{g})\) and obtain the sharp growth result for this family.