Development of inequality and characterization of equality conditions for the numerical radius

Let $A$ be a bounded linear operator on a complex Hilbert space and $\mathrm{\Re }(A)$ ( $\mathrm{\Im }(A)$ ) denote the real part (imaginary part) of A. Among other refinements of the lower bounds for the numerical radius of $A$, we prove that \begin{eqnarray*} w(A)&\geq &\frac{1}{2} \left |A \right| + \frac{ 1}{2} \mid |\Re(A)|-|\Im(A)|\mid,,,\mbox{and}\ w^2(A)&\geq& \frac{1}{4} \left |A^A+AA^ \right| + \frac{1}{2}\mid |\Re(A)|^2-|\Im(A)|^2 \mid, \end{eqnarray*} where $w(A)$ is the numerical radius of the operator $A$. We study the equality conditions for $w(A)=\frac{1}{2}\sqrt{\mathrm{\parallel }{A}^{\ast }A+A{A}^{\ast }\mathrm{\parallel }}$ and prove that $w(A)=\frac{1}{2}\sqrt{\mathrm{\parallel }{A}^{\ast }A+A{A}^{\ast }\mathrm{\parallel }}$ if and only if the numerical range of $A$ is a circular disk with center at the origin and radius $\frac{1}{2}\sqrt{\mathrm{\parallel }{A}^{\ast }A+A{A}^{\ast }\mathrm{\parallel }}$. We also obtain upper bounds for the numerical radius of commutators of operators which improve on the existing ones.