A monotonicity theorem for the family $f_{a}(x)=a-x^{2}$

Abstract:

Let ${f_a}(x) = a - {x^2}$, $x \in [ - \tfrac {1} {2} - \tfrac {1} {2}\sqrt {1 + 4a}$, $\tfrac {1} {2} + \tfrac {1} {2}\sqrt {1 + 4a} ]$ and $a \in [0,2]$. It is proved that if ${f_a}$ has a periodic orbit of odd period $n$ and if $b > a$, then ${f_b}$ has a periodic orbit of period $n$. This is equivalent to the corresponding result for the function family ${g_\lambda }(x) = \lambda x(1 - x)$, $x \in [0,1]$, $\lambda \in [0,4]$.