  In this paper, we are concerned with Jacobi polynomials
$P_n^{(\alpha,\beta)}(x)$ on the Bernstein ellipse with motivation mainly
coming from recent studies of convergence rate of spectral interpolation. An
explicit representation of $P_n^{(\alpha,\beta)}(x)$ is derived in the variable
of parametrization. This formula further allows us to show that the maximum
value of $\left|P_n^{(\alpha,\beta)}(z)\right|$ over the Bernstein ellipse is
attained at one of the endpoints of the major axis if $\alpha+\beta\geq -1$.
For the minimum value, we are able to show that for a large class of Gegenbauer
polynomials (i.e., $\alpha=\beta$), it is attained at two endpoints of the
minor axis. These results particularly extend those previously known only for
some special cases. Moreover, we obtain a more refined asymptotic estimate for
Jacobi polynomials on the Bernstein ellipse.
