Recently, in studing some specific orthogonal polynomials on unit circle, I was lead to study the asymptotic behavior of the following hypergeometric function at the neighberhood of $-1$:

$$ f_n(e^{i \theta}) = _2\!\!F_1(s, -n; -n-s; e^{i\theta}) = \sum_{k = 0}^n \frac{(s)_k(-n)_k}{(-n-s)_k k!} e^{i k \theta}.$$ What is the asymptotic behavior of $ |f_n(e^{i \theta})|$ when $\theta$ is near $\pi$ ? Do we have some estimate as the Szego type inequalities for Jacobi polynomials?

Does some one know any references related to this? Thank you in advance for any kind of suggestions.