Inequalities for functions of \(2\times 2\) block matrices

Let \(T=\left[ \begin{array}{cc} T_{11} &{} T_{12} \\ T_{21} &{} T_{22} \end{array} \right] \) be accretive-dissipative, where \(T_{11},T_{12},T_{21},\) and \(T_{22} \) are \(n\times n\) complex matrices. Let f be a non-negative function on \( [0,\infty )\) such that \(f(0)=0\), and let \(\alpha ,\beta \in (0,1)\) such that \(\alpha +\beta =1\). For every unitarily invariant norm \(\left| \left| \left| \cdot \right| \right| \right| \), it is shown that

$$\begin{aligned} \sum _{j=1}^{2}\left| \left| \left| f\left( \frac{\left| T_{jj}+(2\alpha -1)T_{jj}^{*}\right| }{2\sqrt{2}}\right) +f\left( \sqrt{\frac{\alpha \beta }{2}}\left| T_{jj}^{*}\right| \right) \right| \right| \right| \\ \le 2\max (\alpha ,\beta )\left| \left| \left| f(\left| T\right| )\right| \right| \right| \end{aligned}$$

whenever \(t\rightarrow f\left( \sqrt{t}\right) \) is convex and

$$\begin{aligned} \sum _{j=1}^{2}\left| \left| \left| \alpha f\left( \frac{ \left| T_{jj}+(2\alpha -1)T_{jj}^{*}\right| }{\sqrt{2\alpha }} \right) +\beta f\left( \sqrt{2\alpha }\left| T_{jj}^{*}\right| \right) \right| \right| \right| \\ \le 4\left| \left| \left| f\left( \sqrt{ \max (\alpha ,\beta )}\left| T\right| \right) \right| \right| \right| \end{aligned}$$

whenever f is concave.