Summary: It is known that all solutions of the difference equation \[ \Delta x(n)+p(n)x(n-k)=0, \quad n\geq0, \] where \(\{p(n)\}_{n=0}^{\infty}\) is a nonnegative sequence of reals and \(k\) is a natural number, oscillate if \(\liminf_{n\rightarrow\infty}\sum_{i=n-k}^{n-1}p(i)> (\frac{k}{k+1})^{k+1}\). In the case that \(\sum_{i=n-k}^{n-1}p(i)\) is slowly varying at infinity, it is proved that the above result can be essentially improved by replacing the above condition with \(\limsup_{n\rightarrow\infty}\sum_{i=n-k}^{n-1}p(i)> (\frac{k}{k+1})^{k+1}\). An example illustrating the applicability and importance of the result is presented.