  In this paper we prove the following theorem: Suppose that $f_1,f_2\in
H^\infty_\R(\D)$, with $\norm{f_1}_\infty,\norm{f_2}_{\infty}\leq 1$, with $$
\inf_{z\in\D}(\abs{f_1(z)}+\abs{f_2(z)})=\delta>0. $$ Assume for some
$\epsilon>0$ and small, $f_1$ is positive on the set of $x\in(-1,1)$ where
$\abs{f_2(x)}<\epsilon$ for some $\epsilon>0$ sufficiently small. Then there
exists $g_1, g_1^{-1}, g_2\in H^\infty_\R(\D)$ with $$
\norm{g_1}_\infty,\norm{g_2}_\infty,\norm{g_1^{-1}}_\infty\leq
C(\delta,\epsilon) $$ and $$ f_1(z)g_1(z)+f_2(z)g_2(z)=1\quad\forall z\in\D. $$
