Approximation of univalent functions by compositions (Q1884449)

Let \(S\) be the class of univalent analytic functions \(f\) in the unit disc \(\mathbb D\) with \(f(0)=f^\prime (0)-1=0\). For \(\alpha >1\) and \(\gamma \in \mathbb R\) we let \(p_{\alpha ,\gamma }\) be the conformal mapping from \(\mathbb D\) onto \(\mathbb D\) minus a radial slit ending at \(e^{i\gamma }\) with \(p_{\alpha ,\gamma }(0)=0\) and \(p_{\alpha ,\gamma }^\prime (0)=1/\alpha \). In this paper the author considers the question of approximating a function \(f\in S\) by functions of the type \(\alpha_1\alpha _2\dots \alpha _np_{\alpha _1,\gamma _1} \circ p_{\alpha _2,\gamma _2}\circ \dots \circ p_{\alpha_n,\gamma_n}\). Theorem~1 asserts that if \(f\in S\) and \(n\) and \(m\) are two natural numbers then there exist \(\alpha _j>1\), \(\gamma _j\in \mathbb R\), \(j=1, 2, \dots nm\) such that \[ \begin{aligned} &\max_{| z| =r}\left | f(z)-\alpha_1\alpha _2\dots \alpha _{nm}p_{\alpha _1,\gamma _1} \circ p_{\alpha _2,\gamma _2}\circ \dots \circ p_{\alpha_{nm},\gamma_{nm}}(z)\right | \\ \leq &\frac{11r^2}{(1-r)^6}\frac{\log n}{n} +\frac{15r^2}{(1-r)^5}\frac{1}{m}+ \frac{2r^2}{(1-r)^4}\frac{1}{n},\quad 0<r<1.\end{aligned} \] The proof of this result is based on the representation of a function \(f\in S\) as a solution of the Löwner-Kufarev equation.