Let A be a bounded linear operator on a complex Hilbert space and ℜ(A) (ℑ(A)) denote the real part (imaginary part) of A. Among other refinements of the lower bounds for the numerical radius of A, we prove that [Formula presented] where w(A) and ‖A‖ are the numerical radius and operator norm of A, respectively. We study the equality conditions for [Formula presented] and prove that [Formula presented] if and only if the numerical range of A is a circular disk with center at the origin and radius [Formula presented]. We also obtain upper bounds for the numerical radius of commutators of operators which improve on the existing ones.