Given two holomorphic functions on a region find two other such that...

Solution 1:

This is a special case of Theorem 15.15 in Rudin's Real and Complex Analysis, as Malik Younsi said. I'll simplify the proof for this case.

Our goal is to find a holomorphic function $g_1$ such that $(1-f_1g_1)/f_2$ is holomorphic. Enumerate the distinct zeros of $f_2$ as $\{z_n:n=1,2,\dots\}$ with orders $\{m_n:n=1,2,\dots\}$. We need $1-f_1g_1$ to vanish at every $z_n$ to the order at least $m_n$. Formally, we need $$1-f_1(z)g_1(z) = O((z-z_n)^{m_n+1}), \quad z\to z_n \tag{1}$$ Condition (1) is fulfilled by choosing $g_1$ such that $$ g_1(z) = \frac{1}{f_1(z)} + O((z-z_n)^{m_n+1}), \quad z\to z_n \tag{2} $$ for every $n$. The latter is made possible by the Mittag-Leffler interpolation theorem (note that $1/f_1$ is holomorphic at $z_n$).