On the boundary behaviour of derivatives of functions in the disc algebra (Q721028)

Let \(A(\mathbb D)\) denote the disc algebra of functions analytic on the unit disc \(\mathbb D\) and continuous on \(\overline{\mathbb D}\), and let \((\zeta_k)_{k\geq 1}\) be a sequence of points in \(\partial \mathbb D\). Further, for each \(k\), let \((z_n^k)_{n\geq 1}\) be a sequence of points in \(\mathbb D\) such that \(z_n^k\rightarrow \zeta_k\) as \(n\rightarrow \infty \). A function \(f\in A(\mathbb D)\) is said to belong to the collection \(\mathcal U((\zeta_k),(z_n^k))\) if, for any sequence \((w_k)_{k\geq 1}\) in \(\mathbb C\), there is an increasing sequence \((n_j)\) of natural numbers such that \(f^{(k)}(z_{n_j}^k)\rightarrow w_k\) as \( j\rightarrow \infty \) for every \(k\geq 1\). The main result of this paper says that \(\mathcal U((\zeta_k),(z_n^k))\) is a dense \(G_\delta\)-subset of \(A(\mathbb D)\) and is densely lineable and spaceable.