lp-norms of Fourier coefficients of powers of a Blaschke factor

We determine the asymptotic behavior of the lp-norms of the sequence of Taylor coefficients of bn, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{*{20}{c}} {b(z) = \frac{{z - \lambda }}{{1 - \lambda z}},}&{\left| \lambda \right| < 1,} \end{array}$$\end{document} is an automorphism of the unit disk, p ∈ [1,∞], and n is large. It is known that in the parameter range p ∈ [1, 2] a sharp upper bound \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\left\| {{b^n}} \right\|_{l_p^A}} \leqslant {c_p}{n^{\tfrac{{2 - p}}{{2p}}}}$$\end{document} holds. In this article we find that this estimate is valid even when p ∈ [1, 4). We prove that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\left\| {{b^n}} \right\|_{l_4^A}} \leqslant {c_4}{\left( {\frac{{\log n}}{n}} \right)^{\tfrac{1}{4}}}$$\end{document} and for p ∈ (4,∞] that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\left\| {{b^n}} \right\|_{l_p^A}} \leqslant {c_p}{n^{\tfrac{{1 - p}}{{3p}}}}.$$\end{document} The upper bounds are shown to be asymptotically sharp as n tends to ∞. As a direct application we prove the sharpness of existing upper estimates on analytic capacities in Beurling–Sobolev spaces. Our investigation is also motivated by a question of J. J. Schäffer about optimal estimates for norms of inverse matrices.