On a Cyclic Inequality Related to Chebyshev Polynomials

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 $${\left(\sum _{i=1}^n{x}_i^{(a+b+1)\mathrm{/}2}\right)}^2\ge \sum _{i=1}^n{x}_i\sum _{i=1}^n{x}_i^a{x}_{i+1}^b,$$ where $a,b,x_1,\dots ,x_n$ (with $x_{n+1}=x_1$) are nonnegative. In particular, we prove that the inequality holds when $a=b=1$ and $n\le 8$ for all nonnegative numbers $x_1,\dots ,x_n$.