  We show that if $-A$ generates a bounded $\alpha$-times resolvent family for
some $\alpha \in (0,2]$, then $-A^{\beta}$ generates an analytic $\gamma$-times
resolvent family for $\beta \in(0,\frac{2\pi-\pi\gamma}{2\pi-\pi\alpha})$ and
$\gamma \in (0,2)$. And a generalized subordination principle is derived. In
particular, if $-A$ generates a bounded $\alpha$-times resolvent family for
some $\alpha \in (1,2]$, then $-A^{1/\alpha}$ generates an analytic
$C_0$-semigroup. Such relations are applied to study the solutions of Cauchy
problems of fractional order and first order.
