Lower bound for cyclic sums of Diananda type

Let $C=\inf (k\mathrm{/}n){\sum }_{i=1}^n{x}_i(x_{i+1}+\cdots +x_{i+k}{)}^{-1}$, where the infimum is taken over all pairs of integers $n\ge k\ge 1$ and all positive $x_1,\dots ,x_{n+k}$ subject to cyclicity assumption $x_{n+i}=x_i$, $i=1,\dots ,k$. We prove that $\ln 2\le C<0.9305$. In the definition of the constant $C$ the operation ${\inf }_k{\inf }_n{\inf }_{\mathbf{x}}$ can be replaced by ${\lim }_{k\to \mathrm{\infty }}{\lim }_{n\to \mathrm{\infty }}{\inf }_{\mathbf{x}}$.