Let $T$ be a rectifiable Jordan curve in $\mathbb{C},$ $G$ be the interior of $T,$ and $\Phi$ be a conformal map of the unit disk $\mathbb{D}$ onto $G.$ Let $\mathcal{P}_{n}$ be the space of algebraic polynomials of degree at most $n$ and $\mathcal{M}_n(\mathbb{D})$ be its subset consisting of monic polynomials with $n$ zeros in $\mathbb{D}.$
My question is that for any $v_n\in \mathcal{M}_n(\mathbb{D})$ and $u_{n-1}\in \mathcal{P}_{n-1},$ does there exist a function $g$ that is analytic on the unit disk $\mathbb{D}$ and continuous on the unit circle $\mathbb{S}^1$ such that $$\int_{T} \frac{\left(\frac{u_{n-1}(\Phi^{-1}(\zeta))}{v_n(\Phi^{-1}(\zeta))}+g(\Phi^{-1}(\zeta)) \right)}{q_{n}(\zeta)((\Phi^{-1})'(\zeta))^{1/2}(\zeta-z)}d\zeta=0,\quad \quad \quad z\in G,$$ where $q_n(\zeta):=\prod_{j=1}^n(\zeta-\Phi(\alpha_j))$ and $\alpha_1,\ldots,\alpha_n$ are the zeros of $v_n.$
Could you give a solution to this problem or an idea to the solution? I have not solved a question like this before. I just run out of ideas.