In this article, we present some new general forms of numerical radius inequalities for Hilbert space operators. The significance of these inequalities follow from the way they extend and refine some known results in this field. Among other inequalities, it is shown that if A is a bounded linear operator on a complex Hilbert space, then $w^2\left(A\right)\le ∥{\int }_0^1{\left(t\left|A\right|+\left(1-t\right)\left|A^{\ast }\right|\right)}^2\mathrm{d}t∥\le \frac{1}{2}∥{\left|A\right|}^2+{\left|A^{\ast }\right|}^2∥$ where $w\left(A\right)$ and $∥A∥$ are the numerical radius and the usual operator norm of A, respectively.

In this article, we present new inequalities for the numerical radius of the sum of two Hilbert space operators. These new inequalities will enable us to obtain many generalizations and refinements of some well known inequalities, including multiplicative behavior of the numerical radius and norm bounds. Among many other applications, it is shown that if T is accretive-dissipative, then 1/?2 ||T|| ? ?(T), where ?(?) and ||?||denote the numerical radius and the usual operator norm, respectively. This inequality provides a considerable refinement of the well known inequality 1/2 ||T|| ? ?(T).

In this paper, we discuss and present new sharp inequalities for the numerical radii of Hilbert space operators. In particular, if A and B are bounded linear operators on a Hilbert space, we present new upper bounds for $\omega \left(A^{\ast }B\right)$. The main tool to obtain our results is using block matrix techniques. Among many interesting results, and as an application of the new inequalities, we obtain the following bound for the numerical radius of an operator T, $\omega \left(T\right)\le \frac{1}{2}\left({\Vert T\Vert }^{1/2}\Vert {\left|T\right|}^{1/2}+{\left|T^{\ast }\right|}^{1/2}\Vert \right),$where $\omega (\cdot )$, $\Vert \cdot \Vert$, and $|\cdot |$ denote the numerical radius, the usual operator norm, and the absolute value operator, respectively. Other difference inequalities will be presented too.