Abstract: We derive various lower bounds for the numerical radius w(A) of a bounded linear operator A defined on a complex Hilbert space, which improve the existing inequality w^2(A)geq frac{1}{4}|A^*A+AA^*| . In particular, for rgeq 1 , we show that tfrac{1}{4}|A^*A+AA^*|leqtfrac{1}{2}(tfrac{1}{2}|operatorname{Re}(A)+operatorname{Im}(A)|^{2r}+tfrac{1}{2}|operatorname{Re}(A)-operatorname{Im}(A)|^{2r})^{1/r} leq w^{2}(A), where operatorname{Re}(A) and operatorname{Im}(A) are the real and imaginary parts of A , respectively. Furthermore, we obtain upper bounds for w^2(A) refining the well-known upper estimate w^2(A)leq frac{1}{2}(w(A^2)+|A|^2) . Criteria for w(A)=frac12|A| and for w(A)=frac{1}{2}sqrt{|A^*A+AA^*|} are also given.