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Let $\gamma\subset B(0,2)\subset \mathbb{C}$ a smooth Jordan curve envolving the origin and $\phi:\mathbb{C}\setminus (-\infty,0)\to \mathbb{C}$ an analytic function having finite boundary values, that is, there exist $\phi_+(s)$ and $\phi_-(s)$ for all $s\in(-\infty,0)$.

Fix $z\in\mathbb{C}\setminus \gamma$ and suppose that $$ \left| \int_{\gamma}\frac{e^{n\phi(s)}}{s-z}\ ds\right|\to 0,\quad \text{when}\ n\to\infty. $$

Consider $\{g_n\}$ is a sequence of entire functions satisfying $g_n(z)=O(\frac{1}{n^{\alpha}})$ (in my case this sequence is composed by polynomials of fixed degree vanish at zero), uniformly in $B(0,2)$, for some fixed $0<\alpha<1$, let $\varepsilon>0$ be given. Have I enough hypothesis to say something about $$ \lim_{\varepsilon\to 0}\ \ \limsup_{n\to\infty}\ \ e^{-n\varepsilon}\left| \int_{\gamma}\frac{e^{n(\phi(s)+g_n(s))}}{s-z}\ ds\right| \ ? $$

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