Exactness of a nontrivial estimate in a cyclic inequality

It is proved that the inequality [1]
$$ \frac1n\sum_{i=1}^n\frac{\nu_1a_{i+1}+\nu_2a_{i+2}+\nu_3a_{i+3}}{\delta_2a_{i+2}+\delta_3a_{i+3}}\geqslant\psi(0), $$
where $n\geqslant3$, $\nu_1, \nu_2, \nu_3\geqslant0$, $\delta_2, \delta_3>0$, and $\psi(t)$ is the convex lower support of the function $\widetilde{\psi}(t)$ defined in [1], is exact.