Advanced Refinements of Numerical Radius Inequalities

We prove several numerical radius inequalities for linear operators in Hilbert spaces. It is shown, among other inequalities, that if $A$ is a bounded linear operator on a complex Hilbert space, then $$\omega \left(A\right)\le \frac{1}{2}\sqrt{\parallel {\mid A\mid }^2+{\mid A^{\ast }\mid }^2\parallel +\parallel \mid A\mid \mid A^{\ast }\mid +\mid A^{\ast }\mid \mid A\mid \parallel },$$ where $\omega \left(A\right)$, $\parallel A\parallel$, and $\mid A\mid$ are the numerical radius, the usual operator norm, and the absolute value of $A$, respectively. This inequality provides a refinement of an earlier numerical radius inequality due to Kittaneh, namely, $$\omega \left(A\right)\le \frac{1}{2}(\parallel A\parallel +{\parallel A^2\parallel }^{\frac{1}{2}})\mathrm{.}$$ Some related inequalities are also discussed.