The main result proved here is the inequality $(n + 1)F_n^\alpha (x) nF_n^\beta (x) - nF_{n - 1}^\alpha (x)F_{n - 1}^\beta (x) > 0$ for $ - 1 < x < 1$ and $\frac{1}{2} \leqq \alpha \leqq \beta \leqq \alpha + 1$, where $F_n^\lambda (x) = {{P_n^\lambda (x)} / {P_n^\lambda (1)}}$ and $P_n^\lambda (x)$ is the ultraspherical polynomial. We discuss some other similar inequalities for ultraspherical and Laguerre polynomials.