Norm and Numerical Radius Inequalities for Hilbert Space Operators

In this paper, we present several numerical radius and norm inequalities for sum of Hilbert space operators. These inequalities improve some earlier related inequalities. For A,B\in B\left( H \right), we prove that\[\omega \left( {{B}^{*}}A \right)\le \sqrt{\frac{۱}{۲}{{\left\| A \right\|}^{۲}}{{\left\| B \right\|}^{۲}}+\frac{۱}{۲}\omega \left( {{\left| B \right|}^{۲}}{{\left| A \right|}^{۲}} \right)}\le ۴\omega \left( A \right)\omega \left( B \right).\]