  We present sharp lower bounds for the A-numerical radius of semi-Hilbertian
space operators. We also present an upper bound. Further we compute new upper
bounds for the $B$-numerical radius of $2 \times 2$ operator matrices where $B
= \textit{diag}(A,A)$, $A$ being a positive operator. As an application of the
A-numerical radius inequalities, we obtain a bound for the zeros of a
polynomial which is quite a bit improvement of some famous existing bounds for
the zeros of polynomials.
