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 $x_1,\ldots, x_n$.
Categories: math.CA