We show that if $T$ is a bounded linear operator on a complex Hilbert space, thenbegin{equation*}frac{1}{2}Vert TVertleq sqrt{frac{w^2(T)}{2} + frac{w(T)}{2}sqrt{w^2(T) - c^2(T)}} leq w(T),end{equation*}where $w(cdot)$ and $c(cdot)$ are the numerical radius and the Crawford number, respectively.We then apply it to prove that for each $tin[0, frac{1}{2})$ and natural number $k$,begin{equation*}frac{(1 + 2t)^{frac{1}{2k}}}{{2}^{frac{1}{k}}}m(T)leq w(T),end{equation*}where $m(T)$ denotes the minimum modulus of $T$. Some other related results are also presented.