Norm and numerical radius inequalities for Hilbert space operators

In this paper, we prove some inequalities of the operator norm and the numerical radius for Hilbert spaces operators. More precisely, we prove that if A,B∈B(H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A, B \in \mathbb {B(H)}$$\end{document} and AB=-BA∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$AB=-BA^{*}$$\end{document}, then ω(AB)≤DA‖B‖,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \omega (AB) \le D_{A} \Vert B\Vert , \end{aligned}$$\end{document}where DA=infλ∈CA-λI.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{D}_{A}}=\underset{\lambda \in {\mathbb {C}}}{\mathop {\inf }}\,\left\| A-\lambda I \right\| .$$\end{document}