Sharpening certain cyclic inequalities

This paper studies the lower estimate of cyclic sums of the form
$$\frac1n\sum_{i=1}^n\varphi\left(\ln\frac{a_{i+1}}{a_i},\ln\frac{a_{i+2}}{a_i+1}\right),$$
where $\varphi(x,y)$ is a twice continuous differentiable function on the whole plane, $a_{i+n}=a_i$. A structural description is given of a class of functions $\varphi$ for which the lower bound of this sum is attained for $a_i=\mathrm{const}$, i.e., equal to $\varphi(0,0)$. A means of finding the lower bound in all other cases is indicated. This result sharpens and generalizes a number of well known cyclic inequalities.