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A corollary of Alexander’s inequality. (English) Zbl 1222.46038

For a function \(f\) in a uniform algebra \(A\) on a compact Hausdorff space \(X\), one version of Alexander’s inequality is that
\[ \inf_{g\in A}{\|\bar{f}-g\|}\leq \sqrt{\text{area}(\sigma(f))/\pi}\;, \]
where \(\|\cdot\|\) denotes the sup norm of a function on \(X\), and \(\bar{f}\) denotes the complex conjugate of \(f\).
This note considers the case where \(X\) is the boundary of a compact subset \(Y\) of the plane, the complement of which has a finite number of connected components. Take \(A\) to be the uniform algebra of all functions on \(X\) that can be uniformly approximated by rational functions with poles off of \(Y\), and let \(A_0=\{\,f\in A\,:\,\int_X{f\,dm}=0\,\}\), where \(m\) denotes harmonic measure on \(X\) evaluated at a point in \(Y\). Denote by \({H_0}^1\) the closure of \(A_0\) in \(L^1(X,m)\) and let \(N=\{u\in L^1(X,m)\,:\,\int_X{u(f+\bar{g})\,dm}=0\) \(\forall f,\,g\in A\}\). In this context, the corollary of the paper’s title asserts that, if \(f\) and \(f^{\,\prime}\) are in \(A\), then
\[ \text{dist}_{L^1}\left(\overline{z\cdot f^{\,\prime}\,{{ds}\over{dm}}}\, ,\,{H_0}^1+N\right)\geq {{ \text{dist}(\bar{f},\,A)^2}\over {\|\bar{f}\| }}\;, \] where \(ds\) denotes arc length measure on \(X\).
This reviewer felt that the inclusion of a few more details and connective statements in the proofs might have enhanced the overall readability of the paper.

MSC:

46J10 Banach algebras of continuous functions, function algebras
30H10 Hardy spaces
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