Sharp Inequalities for the Numerical Radii of Block Operator Matrices

In this paper we present several sharp upper bounds for the numerical radii of the diagonal and off-diagonal parts of the 2×2 block operator matrix [ABCD]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left[ {\begin{array}{*{20}{c}} A&B \\ C&D \end{array}} \right]$$\end{document}. Among extensions of some results of Kittaneh et al., it is shown that if T=[A00D]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T = \left[ {\begin{array}{*{20}{c}} A&0 \\ 0&D \end{array}} \right]$$\end{document}, and f and g are non-negative continuous functions on [0,∞) such that f(t)g(t) = t (t ≥ 0), then for all non-negative nondecreasing convex functions h on [0,∞), we obtain that h(wr(T))max(∥1ph(fpr(|A|))+1qh(gqr(|A|))∥,∥1ph(fpr(|D|))+1qh(gqr(|D|))∥)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{*{20}{c}} {h({w^r}(T))} \\ { \leqslant \max \left( {\left\| {\frac{1}{p}h({f^{pr}}(|A|)) + \frac{1}{q}h({g^{qr}}(|A*|))} \right\|,\left\| {\frac{1}{p}h({f^{pr}}(|D|)) + \frac{1}{q}h({g^{qr}}(|D*|))} \right\|} \right)} \end{array}$$\end{document} where p, q > 1 with 1p+1q=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tfrac{1}{p} + \tfrac{1}{q} = 1$$\end{document}, and r min(p, q) ≥ 2.