A product involving the $\beta$-family in stable homotopy theory

In the stable homotopy groups ${\pi }_{q(p^n+p^m+1)-3}(S)$ of the sphere spectrum $S$ localized at the prime $p$ greater than three, J. Lin constructed an essential family ${\xi }_{m,n}$ for $n\ge m+2>5$. In this paper, the authors show that the composite ${\xi }_{m,n}{\beta }_s\in {\pi }_{q(p^n+p^m+sp+s)-5}(S)$ for $2\le s<p$ is non-trivial, where $q=2(p-1)$ and ${\beta }_s\in {\pi }_{q(sp+s-1)-2}(S)$ is the known $\beta$-family. We show our result by explicit combinatorial analysis of the (modified) May spectral sequence.