On the existence of short trajectories of quadratic differentials related to generalized Jacobi polynomials with non real varying parameters

The study of the asymptotic distributions of zeros of generalized Jacobi polynomials with non real varying parameters, leads with quadratic differentials. In fact, the support of the limit measure of the root-counting measures sits on the finite critical trajectories of a related quadratic differential. In this paper, we study the trajectories of this quadratic differential, more precisely, we give a necessary and sufficient condition on the complex numbers a; b; and {\lambda} for the existence of at list one finite critical trajectory of the quadratic differential (({\lambda}^2(z-a)(z-b))/((z^2-1)^2))dz^2.