Parametric representation of univalent functions with boundary regular fixed points

Given a set $\mathfrak{S}$ of conformal maps of the unit disk $\mathbb{D}$ into itself that is closed under composition, we address the question whether $\mathfrak{S}$ can be represented as the reachable set of a Loewner - Kufarev - type ODE $\mathrm{d}{w}_t\mathrm{/}\mathrm{d}t=G_t\circ w_t$, $w_0=$\mathsf{i}\mathsf{d}${\mathbb D}$,wherethecontrolfunctions$t\mapsto G_t$formaconvexcone$\mathcal M{\mathfrak S}$\mathrm{.}Forthesetofallconformal$\varphi:\mathbb D\to\mathbb D$with$\varphi(0)=0$,$\varphi’(0)>0$, an affirmative answer to this question is the essence of Loewner’s well-known Parametric Representation of univalent functions [Math. Ann. 89 (1923), 103-121]. In this paper, we study classes of conformal self-maps defined by their boundary regular fixed points and, in part of the cases, establish their Loewner-type representability.