For polynomials $P(z) = c\prod _{k = 1}^n (z - a_k )$ with $a_1 ,a_2 , \cdots ,a_n $ real, we write $P| {(x + iy)} |^2 = \sum _{k = 0}^n L_k (P;x)y^{2k} $, where \[ L_k (P;x) = \sum_{j = 0}^{2k} {\frac{{( - 1)^{j + k} }}{{(2k)!}}} \left( {\begin{array}{*{20}c} {2k} \\ j \\ \end{array} } \right)P^{(j)} (x)P^{(2k - j)} (x).\] We show for $k = 1,2$ and 3 that the functions $L_k (P^{(m)} ;x)$, $m \geqq 0$, for Jacobi polynomials $P(x) = P_n^{(\alpha ,\beta )} (x)$ and their derivatives satisfy the inequalities $L_k (P^{(m)} ;x) \leqq L_k (P^{(m)} ;1)$ for $ - 1 < x < 1 $ and $ - 1 < \alpha = \beta $; $L_k (P^{(m)} ;x) \leqq L_k (P^{(m)} ;1)$ for $0 \leqq x \leqq 1$ and $ - 1 < \beta < \alpha $; and $L_k (P^{(m)} ;x) \leqq L_k (P^{(m)} ; - 1)$ for $ - 1 \leqq x \leqq 0$ and $ - 1 < \alpha < \beta $.