Continuity in the uniform operator topology of a map

Question

I have a question concerning the continuity for $t>0$ in the uniform operator topology $L(X)$ of the following map: $$t\mapsto A^\alpha R(t)$$ where A is the infinitesimal generator of an analytic semigroup, $0<\alpha<1$ and $R(t)$ is the analytic resolvent operator introduced by Grimmer for integrodifferential equations. Actually, I found it as fact in many papers but when I tried the prove it I found difficulty. Please help.