On a class of functions of bounded variation on the line defined by their values on a half-line

Let $\mathscr L$ be the class of functions analytic in the half-plane $\{\operatorname{Im}t>0\}$ and continuous in $\{\operatorname{Im}t\ge0\}$, representable as Fourier transforms of finite complex measures $\mu$, $\operatorname{supp}\mu\subset\mathbb R$, $-\infty\in\operatorname{supp}\mu$ and nonvanishing in $\{\operatorname{Im}t>0\}$; let $\mathscr L_1$ be the linear envelope of $\mathscr L$. It is proved (theorem 1) that
$$ H_i\in\mathscr L_1,(i=1,2),H_1(x)=H_2(x)\text{ for }x<0\Longrightarrow H_1\equiv H_2. $$
This uniqueness theorem is deduced from the following generalization of the Schottky–Landau theorem (theorem 2): let $g_1,\dots,g_p$ be nonvanishing functions analytic in the disc $\{|z|<1\}$ and lizearly independent over $\mathbb C$. Then $|g_k(z)|\le\exp(A(1-|z|)^{-1})(|z|<1,k=1,\dots,p, A\quad\text{not depending on}\quad z)$ provided $\sum_{k=1}^pg_k$ is bounded in the unit disc.