Let a family of univalent functions \(f(z,t):\mathbb{D}\to\mathbb{C}\), \(t\geq 0\), be a Löwner chain satisfying \(\dot f(z,t)=zf'(z,t)p(z,t)\) for \(z\) in the unit disc \(\mathbb{D}\), a.e. \(t\geq 0\), where \(p(z,t)\) is analytic for \(z\in\mathbb{D}\) and measurable for \(t\geq 0\) with \({\mathfrak R}[p(z,t)]\geq 0\). The symbols \(\dot f(z,t)\) and \(f'(z,t)\) denote the derivatives of \(f(z,t)\) w.r.t. \(t\) and \(z\) respectively. Assume further that \(f(z,t)\) is a \(k\)-chain, that is, \(|[p(z,t)-1]/[p(z,t)+1)]|\leq k\) for \(z\in\mathbb{D}\), a.e. \(t\geq 0\), and \(k0\) depends only on \(k\) and \(H^ p\) denotes the usual Hardy space. An example due to Ch. Pommerenke is provided which shows the above does not hold if \(k=1\). Further examples of Löwner chains are constructed using Bazilevič functions. The author remarks that if \(f(z,t)\) is a Löwner chain of Bazilevič type, then \(\log f'(\cdot,t)\in BMOA\) for all \(t\geq 0\), and if \(f(z,t)\) is further assumed to be a \(k\)-chain, for \(k<1\), then \(\log f'(\cdot,t)\in H^ \infty\) for all \(t\geq 0\).