OICC Press International Journal of Mathematical Modelling & Computations 2228-6233 12 3 2022 09 01 Norm and Numerical Radius Inequalities for Hilbert Space Operators 173 181 10.30495/ijm2c.2022.1932741.1214 EN Mohsen Omidvar Department of Mathematics, Mashhad Branch, Islamic Azad University Mahdi Ghasvareh Department of Mathematics, Mashhad Branch, Islamic Azad University Journal Article 2022 09 01 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\[\omega \left( {{B}^{*}}A \right)\le \sqrt{\frac{1}{2}{{\left\| A \right\|}^{2}}{{\left\| B \right\|}^{2}}+\frac{1}{2}\omega \left( {{\left| B \right|}^{2}}{{\left| A \right|}^{2}} \right)}\le 4\omega \left( A \right)\omega \left( B \right).\]