We study a frequently investigated class of linear difference equations $\Delta v\left(n\right)=-p\left(n\right)v\left(n-k\right)$ with a positive coefficient $p\left(n\right)$ and a single delay *k*. Recently, it was proved that if the function $p\left(n\right)$ is bounded above by a certain function, then there exists a positive vanishing solution of the considered equation, and the upper bound was found. Here we improve this result by finding even the lower bound for the positive solution, supposing the function $p\left(n\right)$ is bounded above and below by certain functions.

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