We have $$x^3+(m+n+p-1)x^2-((m+n)(1-p)+2p-1-mn)x-(p-1)(m-1)(n-1)=0$$ in which $m,n\ge2, p\ge1$ are natural numbers. All the three roots of this cubic are positive. Let $\lambda$ be the least of them.
How $\lambda$ changes with respect to $m,n$? (keeping $m+n,p$ fixed)
For given values of $p$, how to compare corresponding $\lambda$'s?