This paper presents several numerical radii and norm inequalities for Hilbert space operators. These inequalities improve some earlier related inequalities. For an operator $A$, we prove that $$\begin{array}{ccc}\hfill {\omega }^2\left(A\right)\le & \parallel \frac{A^{*}A+A{A}^{*}}{2}-\frac{1}{2R}((1-t)A^{*}A+tA{A}^{*}\hfill & \hfill -{((1-t){\left(A^{*}A\right)}^{1/2}+{\left(A{A}^{*}\right)}^{1/2})}^2)\parallel \end{array}$$ where $R=max\{t,1-t\}$ and $0\le t\le 1$.