Lower bound for cyclic sums of Diananda type

Abstract

Let ${C={\rm inf} (k/n)\sum_{i=1}^n x_i(x_{i+1}+\cdots+x_{i+k})^{-1}}$, where the infimum is taken over all pairs of integers ${n\geq k\geq 1}$ and all positive ${x_1,\ldots,x_n}$, ${x_{n+i}=x_i}$. We prove that ${\ln 2 \leq C < 0.9305}$. In the definition of the constant C, the operation ${{\rm inf}_{k}\, {\rm inf}_{n}\, {\rm inf}_{x}}$ can be replaced by ${{\rm lim}_{k \to \infty}\, {\rm lim}_{n \to \infty} {\rm inf}_{x}}$.