  It is shown that if the dual of a Banach space satisfies Property($K^*$) then
$R(X) \leq 1 + \frac{\displaystyle 1}{\displaystyle 1 + r_{X^*}(1)} < 2$ where
$r_{X^*}(c)$ is Opial's modulus for $X^*.$ Thus $X$ has the weak fixed point
property.
