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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 pointproperty.

Property($K^*$) Implies $R(X) \leq 1 + \frac{\displaystyle 1}{\displaystyle 1 + r_{X^*}(1)}$

Property($K^$) Implies $R(X) \leq 1 + \frac{\displaystyle 1}{\displaystyle 1 + r_{X^}(1)}$