When will a One Parameter Family of Unimodal Maps Produce Finite Limit Cycles Monotonically with the Parameter?

In this note we consider a collection $\cal{C}$ of one parameter families of unimodal maps of $[0,1].$ Each family in the collection has the form $\{\mu f\}$ where $\mu\in [0,1].$ Denoting the kneading sequence of $\mu f$ by $K(\mu f)$, we will prove that for each member of $\cal{C}$, the map $\mu\mapsto K(\mu f)$ is monotone. It then follows that for each member of $\cal{C}$ the map $\mu\mapsto h(\mu f)$ is monotone, where $h(\u{f})$ is the topological entropy of $\mu f.$ For interest, $\mu f(x)=4\mu x(1-x)$ and $\mu f(x)=\mu\sin(\pi x)$ are shown to belong to $\cal{C}.$ This extends the work of Masato Tsujii [1].