relations and provides the bound $$\|a\|\leq\|\Re c\|+\|\Im c\|
\left(\frac{r^2+2+r\sqrt{r^2+4}}{2}+r\sqrt{\frac{r^2+2+r\sqrt{r^2+4}}{2}}\right),$$ as claimed in our remark.

Relation is proved by a direct application of one of the equivalent definitions of positivity in a von Neumann algebra and the Cauchy-Buniakovsky-Schwarz inequality in Hilbert spaces. Let $\xi$ be an arbitrary vector in the Hilbert space $\mathcal H^n$ on which $M_n(
\mathcal A)$ acts as a von Neumann algebra. As we have seen in the proof of above, $a\in B^+_n(c,r)\iff (a-c)(\Im c)^{-1}(a-c)^*\leq r^2\Im a$. This means that $$\left\langle (a-c)(\Im c)^{-1}(a-c)^*\xi,\xi\right\rangle\leq r^2\langle\Im a\xi,\xi\rangle.$$ Moving $a-c$ to the right with a star and taking real and imaginary parts provides us with $$\begin{aligned}
\lefteqn{\left\|(\Im c)^{-\frac12}\Re(a-c)\xi\right\|_2^2+\left\|(\Im c)^{-\frac12}\Im a\xi\right\|_2^2
+\langle\Im c\xi,\xi\rangle}\\
& & \mbox{}+i\left(\left\langle(\Im c)^{-\frac12}\Re(a-c)\xi,(\Im c)^{-\frac12}\Im a\xi\right\rangle
-\overline{\left\langle(\Im c)^{-\frac12}\Re(a-c)\xi,(\Im c)^{-\frac12}\Im a\xi\right\rangle}\right)\\
&\leq &(2+r^2)\langle\Im a\xi,\xi\rangle.\end{aligned}$$ Second line above is simply $-2\Im\left\langle(\Im c)^{-\frac12}\Re(a-c)\xi,(\Im c)^{-\frac12}\Im 
a\xi\right\rangle$, which is clearly greater than $-2\left|
\left\langle(\Im c)^{-\frac12}\Re(a-c)\xi,(\Im c)^{-\frac12}\Im a\xi\right\rangle\right|.$ By the Schwarz-Cauchy inequality (applied in the second inequality below) we obtain $$\begin{aligned}
\langle\Im c\xi,\xi\rangle & \leq & \langle\Im c\xi,\xi\rangle+\left(
\left\|(\Im c)^{-\frac12}\Re(a-c)\xi\right\|_2-\left\|(\Im c)^{-\frac12}\Im a\xi\right\|_2\right)^2\\
& = & \langle\Im c\xi,\xi\rangle+\left\|(\Im c)^{-\frac12}\Re(a-c)\xi\right\|_2^2+\left\|(\Im 
c)^{-\frac12}\Im a\xi\right\|_2^2\\
& & \mbox{}-2\left\|(\Im c)^{-\frac12}\Re(a-c)\xi\right\|_2\left\|(\Im c)^{-\frac12}\Im a\xi\right\|_2\\
& \leq & \langle\Im c\xi,\xi\rangle+\left\|(\Im c)^{-\frac12}\Re(a-c)\xi\right\|_2^2+\left\|(\Im 
c)^{-\frac12}\Im a\xi\right\|_2^2\\
& & \mbox{}-2\left|
\left\langle(\Im c)^{-\frac12}\Re(a-c)\xi,(\Im c)^{-\frac12}\Im a\xi\right\rangle\right|\\
& \leq & \langle\Im 