## 5 Numerical Radius of Real and Imaginary Parts of \(C(p)\)
\begin{array}{c|cccc}-\text{Re}(a_{2n})&\frac{-a_{2n-1}+1}{2}& \ldots&\frac{-a_{n+2}}{2}&-\frac{a_{n+1}}{2}\\ \hline\frac{-\overline{a_{2n}-1}+1}{2}&0&\frac{1}{2}&0&\cdots&0\\ \vdots&\frac{1}{2}&0&\frac{1}{2}&0&\vdots\\ \vdots&0&\frac{1}{2}&\ddots&\ddots&0\\ \frac{-\overline{a_{n+1}}}{2}&\vdots&\ddots&\ddots&\ddots&\frac{1}{2}\\ 0&\cdots&0&\frac{1}{2}&0\end{array} \right],\]  
which can be written as  
\[P_{11}=\left[ \begin{array}{cc}-\text{Re}(a_{2n})&u^{*}\\ u&T_{n-1}\end{array} \right],\]  
where \(x:=\Big{[}\frac{-\overline{p_{2n}-1}+1}{2},\cdots,\frac{-\overline{n_{n+2}}}{ 2},\frac{-\overline{n_{n+1}}}{2}\Big{]}\). Thus, by employing Lemmas 2 and 3, we have  
\[\omega(P_{11}) =\omega\Big{(}\left[ \begin{array}{cc}|\text{Re}(a_{2n})|&x^{*}\\ x&T_{n-1}\end{array} \right]\Big{)}\leq\omega\Big{(}\left[ \begin{array}{cc} \omega(|\text{Re}(a_{2n})|)&\|x\|\\ \|x\|&\omega(T_{n-1})\end{array} \right]\Big{)}\] \[=r\bigg{(}\left[ \begin{array}{cc}\omega(|\text{Re}(a_{2n})|)&\| x\|\\ \|x\|&\omega(T_{n-1})\end{array} \right]\bigg{)}\] \[=\frac{1}{2}\Bigg{(}|\text{Re}(a_{2n})|+\cos\Big{(}\frac{\pi}{n} \Big{)}+\sqrt{\big{(}|\text{Re}(a_{2n})|-\cos\Big{(}\frac{\pi}{n}\Big{)} \Big{)}^{2}+4\|x\|^{2}}\Bigg{)}\] \[=\frac{1}{2}\Bigg{(}|\text{Re}(a_{2n})|+\cos\Big{(}\frac{\pi}{n} \Big{)}+\sqrt{\Big{(}|\text{Re}(a_{2n})|-\cos\Big{(}\frac{\pi}{n}\Big{)} \Big{)}^{2}+|1-a_{2n-1}|^{2}+\sum_{k=n+1}^{2n-2}|a_{k}|^{2}}\Bigg{)}. \tag{21}\]  
On the other hand, we have \[P_{12}+P_{21} =\left[ \begin{array}{ccccc}\frac{-\frac{a_{n}-\frac{a_{n-1}}{2}}}{2}& \ldots&\frac{-a_{2}}{2}&\frac{-a_{1}}{2}\\ 0&0&\ldots&0&0\\ 0&0&\ldots&0&0\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ \frac{1}{2}&0&\cdots&0&0\end{array} \right]+\left[ \begin{array}{ccccc}\frac{- \frac{a_{n}}{2}}{2}&0&\cdots&0&\frac{1}{2}\\ \frac{-\frac{a_{n-1}}{2}}{2}&0&\cdots&\cdots&0\\ \vdots&0&\ddots&\cdots&0\\ \frac{-\frac{a_{n}}{2}}{2}&\vdots&\cdots&\cdots&\vdots\\ \frac{-\frac{a_{n}}{2}}{2}&0&\cdots&0&0\end{array} \right]\] \[=\left[ \begin{array}{ccccc}-\frac{-\frac{a_{n}+\frac{a_{n}}{2}} }{2}&-\frac{a_{n-1}}{2}&\cdots&-\frac{a_{2}}{2}&-\frac{a_{1}+\frac{1}{2}}{2}\\ \hline-\frac{\frac{a_{n-1}}{2}}{2}&0&0&\cdots&0\\ \vdots&\vdots&\ddots&\cdots&0\\ -\frac{\frac{a_{n}}{2}}{2}&0&\cdots&\cdots&\vdots\\ -\frac{\frac{a_{n}}{2}}{2}+\frac{1}{2}&0&0&\cdots&0\end{array} \right]_{n\times n}.\]  
Let \(v:=\left[\frac{-\frac{a_{n-1}}{2}}{2},\cdots,\frac{-\frac{a_{n}}{2}}{2},\frac {-\frac{a_{1}}{2}+1}{2}\right]^{T}\). So that, by Lemmas 2 and 3, we have  
\[\omega(P_{12}+P_{21}) =\omega\bigg{(}\left[ \begin{array}{cc}\left|\mathrm{Re}(a_{n}) \right|&v^{*}\\ v&0\end{array} \right]\bigg{)}\leq\omega\bigg{(}\left[ \begin{array}{cc}w( \left|\mathrm{Re}(a_{n})\right|)&\left\|v\right\|\\ \left\|v\right\|&0\end{array} \right]\bigg{)}\] \[=r\bigg{(}\left[ \begin{array}{cc}w(\left|\mathrm{Re}(a_{n}) \right|)&\left\|v\right\|\\ \left\|v\right\|&0\end{array} \right]\bigg{)}\] \[=\frac{1}{2}\bigg{(}\left|\mathrm{Re}(a_{n})\right|+\sqrt{\left| \mathrm{Re}(a_{n})\right|^{2}+4\left\|v\right\|^{2}}\bigg{)}\] \[=\frac{1}{2}\Bigg{(}\left|\mathrm{Re}(a_{n})\right|+\sqrt{\left| \mathrm{Re}(a_{n})\right|^{2}+\left|1-a_{1}\right|^{2}+\sum_{k=2}^{n-1}\left| a_{k}\right|^{2}}\bigg{)}\,. \tag{22}\]  
Similarly, we have  
\[P_{12}-P_{21}=\left[ \begin{array}{ccccc}-\frac{a_{n}-\frac{a_{n}}{2}}{2}&- \frac{a_{n-1}}{2}&\cdots&-\frac{a_{2}}{2}&-\frac{a_{1}}{2}-\frac{1}{2}\\ \hline\frac{a_{n-1}}{2}&0&0&\cdots&0\\ \vdots&\vdots&\ddots&\cdots&0\\ \frac{a_{n}}{2}&0&\cdots&\cdots&\vdots\\ \frac{a_{n}}{2}+\frac{1}{2}&0&0&\cdots&0\end{array} \right]_{n\times n}.\]  
Let \(u:=\left[\frac{\frac{a_{n-1}}{2}}{2},\cdots,\frac{\frac{a_{n}}{2}}{2},\frac {\frac{a_{1}}{2}+1}{2}\right]^{T}\). So that, by Lemmas 2 and 3, we have  
\[\omega(P_{12}-P_{21}) =\omega\bigg{(}\left[ \begin{array}{cc}\left|-i\mathrm{Im}(a_{n} )\right|&u^{*}\\ u&0\end{array} \right]\bigg{)}\leq\omega\bigg{(}\left[ \begin{array}{cc}w( \left|\mathrm{Im}(a_{n})\right|)&\left\|u\right\|\\ \left\|u\right\|&0\end{array} \right]\bigg{)}\] \[=r\bigg{(}\left[