Loewner–Kufarev ODE $$\label{EQ_LK-ODE0}
\frac{{\mathrm{d}}w}{{\mathrm{d}}t}=-w\,p(w,t),\quad t\ge0,\qquad w(z,0)=z\in{\mathbb{D}},$$ has a unique solution $w=w(z,t)$ and the locally uniform limit $$\label{EQ_f-limit}
f(z):=\lim_{t\to+\infty}\frac{\,\mathrlap{w(z,t)}\hphantom{w'(0,t)}}{\,w'(0,t)},\quad z\in{\mathbb{D}},$$ where $w'$ denotes the derivative w.r.t. $z$, exists and belongs to ${\mathcal{S}}$. On the other hand, see e.g. [@Pommerenke Theorem 6.1 on p.159] or [@Gut], *every* function $f\in{\mathcal{S}}$ can be represented by  with a suitable, and in general not unique, *normalized* Herglotz function, i.e. a Herglotz function $p$ with $\Re p(0,t)=1$ for a.e. $t\ge0$.

A natural problem arises: *given a subclass ${\mathrlap{\hskip.125em\widetilde{\hphantom{{\mathcal{S}}}\vphantom{\vbox to1.5ex{\vss}}}}{\mathcal{S}}\vphantom{\vbox to2.3ex{\vss}}}\subset{\mathcal{S}}$, find a class of Herglotz functions that generates ${\mathrlap{\hskip.125em\widetilde{\hphantom{{\mathcal{S}}}\vphantom{\vbox to1.5ex{\vss}}}}{\mathcal{S}}\vphantom{\vbox to2.3ex{\vss}}}$ via .* The answer is known in some cases, e.g. for starlike functions, bounded univalent functions, and for univalent functions with real Taylor coefficients;

A partial answer is also known for the subclass ${\mathcal{S}}_k$, $k\in(0,1)$, formed by all $f\in{\mathcal{S}}$ admitting $k$-q.c. extension to ${\mathbb{C}}$. Namely, in 1972, Becker [@Becker72] found a condition on $p$ in the Loewner–Kufarev equation , see Sect.\[SS\_Becker\], such that the function $f$ given by  belongs to ${\mathcal{S}}_k$. The class ${\mathcal{S}}_k^B$ generated by Herglotz functions that satisfy Becker’s condition is a *proper* subset of ${\mathcal{S}}_k$. In this paper we study ${\mathcal{S}}_k^B$ and its relation with ${\mathcal{S}}_k$. In particular, in Sect.\[SS\_a3\] we find the 