The estimate will often be used under the equivalent forms $$\begin{aligned}
\lefteqn{\left[(\Im f(a))^{-\frac12}\Delta f(a,c)(b)(\Im f(c))^{-\frac12}\right]^*
\left[(\Im f(a))^{-\frac12}\Delta f(a,c)(b)(\Im f(c))^{-\frac12}\right]}\nonumber\\ & \leq & 
\left\|\left(\Im a\right)^{-\frac12}b\left(\Im c\right)^{-\frac12}\right\|_\mathcal A^2\cdot
1_{M_n(\mathcal C)},\label{est}
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\end{aligned}$$ $$\begin{aligned}
\lefteqn{\left[(\Im f(a))^{-\frac12}\Delta f(a,c)(b)(\Im f(c))^{-\frac12}\right]
\left[(\Im f(a))^{-\frac12}\Delta f(a,c)(b)(\Im f(c))^{-\frac12}\right]^*}\nonumber\\ & \leq & 
\left\|\left(\Im a\right)^{-\frac12}b\left(\Im c\right)^{-\frac12}\right\|_\mathcal A^2\cdot
1_{M_n(\mathcal C)},\label{est'}
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\end{aligned}$$ $$\Delta f(a,c)(b)^*(\Im f(a))^{-1}
\Delta f(a,c)(b)\leq
\left\|\left(\Im a\right)^{-\frac12}b\left(\Im c\right)^{-\frac12}\right\|_\mathcal A^2\cdot
\Im f(c),\label{est''}$$ $$\Delta f(a,c)(b)(\Im f(c))^{-1}
\Delta f(a,c)(b)^*\leq
\left\|\left(\Im a\right)^{-\frac12}b\left(\Im c\right)^{-\frac12}\right\|_\mathcal A^2\cdot
\Im f(a),\label{est'''}$$ which we give here for convenience. Of course, if $b\in M_n(\mathcal A)$, the notation $\|b\|_\mathcal A$ signifies the C${}^*$-norm of $b$ as an element in $M_n(\mathcal A)$.

As $\Im \left(\begin{array}{cc}
a & b\\
0 & c
\end{array}\right)=
\left(\begin{array}{cc}
\Im a & \frac{b}{2i}\\
\left(\frac{b}{2i}\right)^* & \Im c
\end{array}\right)$, we have $\left(\begin{array}{cc}
a & b\\
0 & c
\end{array}\right)\in H^+(M_{2n}(\mathcal A))$ if and only if $a,c\in H^+(M_{n}(\mathcal A))$ and $b^*(\Im a)^{-1}b<4\Im c$. This last relation is equivalent to $\left[(\Im a)^{-\frac12}b
(\Im c)^{-\frac12}\right]^*\left[(\Im a)^{-\frac12}b
(\Im c)^{-\frac12}\right]<4$, or $\|(\Im a)^{-\frac12}b
(\Im c)^{-\frac12}\|_\mathcal A<2$. Thus, as $f$ maps the noncommutative upper half-plane into itself, and for any $b_0\in M_n(\mathcal A)$ there exists an $\varepsilon_{b_0}=\frac{2}{\|(\Im a)^{-\frac12}b_0
(\Im c)^{-\frac12}\|_\mathcal A}>0$ such that $$\left(\begin{array}{cc}
a & \varepsilon {b_0}\\
0 & c
\end{array}\right)\in H^+(M_{2n}(\mathcal A))\quad\text{for all }
\varepsilon\in[0,\varepsilon_{b_0}),$$ and 