Stabilization in $H_{\mathbb{R}}^{\mathrm{\infty }}(\mathbb{D})$

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$${\inf }_{z\in \text{D}}(\text{abs}{f}_1(z)+\text{abs}{f}_2(z))=\delta >0.$$Assumeforsome$\epsilon>0$andsmall,$f_1$ispositiveonthesetof$x\in(-1,1)$where$\abs{f_2(x)}<\epsilon$forsome$\epsilon>0$ sufficiently small. Then there exists $g_1,g_1^{-1},g_2\in H_{\mathbb{R}}^{\mathrm{\infty }}(\text{D})$ with $$\text{norm}{g_1}_{\mathrm{\infty }},\text{norm}{g_2}_{\mathrm{\infty }},\text{norm}{g_1^{-1}}_{\mathrm{\infty }}\le C(\delta ,ϵ)$$ and $$f_1(z)g_1(z)+f_2(z)g_2(z)=1\mathrm{\forall }z\in \text{D}\mathrm{.}$$