Lower bound for cyclic sums of Diananda type

TL;DR

This paper establishes bounds for a cyclic sum constant related to Diananda's problem, proving it is at least ln(2) and less than 0.9305, with implications for the asymptotic behavior of these sums.

Contribution

The authors derive new bounds for the cyclic sums of Diananda type, improving understanding of their asymptotic behavior and providing a rigorous proof of the bounds.

Findings

Lower bound for C is ln(2).

Upper bound for C is 0.9305.

Limits of the infimum are interchangeable with limits as k and n go to infinity.

Abstract

Let $C=\inf (k/n)\sum_{i=1}^n x_i(x_{i+1}+\dots+x_{i+k})^{-1}$, where the infimum is taken over all pairs of integers $n\geq k\geq 1$ 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 $\ln 2\leq C< 0.9305$. In the definition of the constant $C$ the operation $\inf_k\inf_n\inf_{\mathbf{x}}$ can be replaced by $\lim_{k\to\infty}\lim_{n\to\infty}\inf_{\mathbf{x}}$.