so $$\left(\begin{array}{cc}
f(a) & \varepsilon\Delta f(a,c)({b_0})\\
0 & f(c)
\end{array}\right)\in H^+(M_{2n}(\mathcal A))\quad\text{for all }
\varepsilon\in[0,\varepsilon_{b_0}).$$ In particular $\varepsilon\left\|\left(\Im f(a)\right)^{-\frac12}
\Delta f(a,c)({b_0})\left(\Im f(c)\right)^{-\frac12}\right\|_\mathcal C<2$ for $\varepsilon<\frac{2}{\|(\Im a)^{-\frac12}b_0
(\Im c)^{-\frac12}\|_\mathcal A}$. Letting $\varepsilon\to\frac{2}{\|(\Im a)^{-\frac12}b_0
(\Im c)^{-\frac12}\|_\mathcal A}$ from below, we obtain $$\left\|\left(\Im f(a)\right)^{-\frac12}
\Delta f(a,c)({b_0})\left(\Im f(c)\right)^{-\frac12}\right\|_\mathcal C\leq
\|(\Im a)^{-\frac12}b_0(\Im c)^{-\frac12}\|_\mathcal A.$$ As $b_0\in M_n(\mathcal A)$ has been chosen arbitrarily, we can replace it by $(\Im a)^{\frac12}b(\Im c)^{\frac12}$ to conclude that, as claimed $$\left\|\left(\Im f(a)\right)^{-\frac12}\Delta f
(a,c)\left((\Im a)^{\frac12}b(\Im c)^\frac12\right)
\left(\Im f(c)\right)^{-\frac12}\right\|_\mathcal C\leq\|b\|_\mathcal A,\quad b\in
M_n(\mathcal A).$$

Clearly, the same method can be used to obtain estimates involving $\Delta^jf$ for all $j\in\mathbb N$. We give one such estimate pertaining to a special case of $j=2$. We shall simply apply the above result to appropriately chosen elements in $M_2(\mathcal A)$. Let $$\left(\begin{array}{cccc}
a_1 & 0 & 0 & 0 \\
0 & a_2 & c & 0 \\
0 & 0 & a_3 & b \\
0 & 0 & 0 & a_4 
\end{array}\right)$$ be such that $\Im a_j>0$ and $
\left(\begin{array}{cc}
a_3 & b\\
0 & a_4
\end{array}\right)\in H^+(M_2(\mathcal A))$. From [@ncfound Section 3] we obtain $$\begin{aligned}
\lefteqn{f\left(\begin{array}{cccc}
a_1 & 0 & 0 & 0 \\
0 & a_2 & c & 0 \\
