  We show that any weighted geometric mean of Chebyshev polynomials is bounded
from above by another Chebyshev polynomial. We also study a related homogeneous
cyclic inequality $$ \left (\sum_{i=1}^n x_i^{(a+b+1)/2} \right )^2 \geq
\sum_{i=1}^n x_i \sum_{i=1}^n x_i^a x_{i+1}^b,$$ where $a,b,x_1,\ldots, x_n$
(with $x_{n+1}=x_1$) are nonnegative. In particular, we prove that the
inequality holds when $a=b=1$ and $n\leq 8$ for all nonnegative numbers
$x_1,\ldots, x_n$.
