Reducibility of function pairs in \(H^\infty_{\mathbb R}\) (Q2849081)

Let \(\mathbb{D}\) denote the unit disc, \(H^\infty\) denote the space of bounded analytic functions on \(\mathbb{D}\), and let \(\left\| f\right\|_{\infty}\) denote the sup norm of \(f\) on \(\mathbb{D}\). The space \(H^\infty_{\mathbb{R}}\) is the set of bounded analytic functions on \(\mathbb{D}\) that are real on the real axis. NEWLINENEWLINENEWLINENEWLINE The reviewer showed that, if \(f,g\in H^\infty_\mathbb{R}\) are functions with \(\left\| f\right\|_{\infty}\) and \(\left\| g\right\|_{\infty}\leq 1\) and \(\delta<\left| f(z)\right|+\left| g(z)\right|\) for all \(z\in\mathbb{D}\) and if there exists an \(\epsilon>0\) such that \(f\) has constant sign on the set \(\{x\in(-1,1): \left| g(x)\right|<\epsilon\}\), then there exists a finite constant \(C(\delta,\epsilon)\) such that the equation \(uf+gh=1\) has a solution \(u,h\in H^\infty_{\mathbb{R}}\) with \(u\) invertible (in \(H^\infty_{\mathbb{R}}\)) and NEWLINE\[NEWLINE \left\| u\right\|_{\infty}+\left\| u^{-1}\right\|_{\infty}+\left\| h\right\|_{\infty}\leq C(\delta,\epsilon). NEWLINE\]NEWLINENEWLINENEWLINEThe author of the paper under review gives a very clever example to show that a sign condition on \((-1,1)\) alone is not sufficient to guarantee the invertibility of the element \(u\). He also provides another, much easier, proof of the result mentioned above.