  Let $\mathcal{A}$ be a unital C*-algebra. Then the theory of Hilbert
C*-modules tells that \begin{align*} \sum_{i=1}^{n}(a_ia_i^*)^\frac{1}{2}\leq
\sqrt{n} \left(\sum_{i=1}^{n}a_ia_i^*\right)^\frac{1}{2}, \quad \forall n \in
\mathbb{N}, \forall a_1, \dots, a_n \in \mathcal{A}. \end{align*} By
modifications of arguments of Botelho-Andrade, Casazza, Cheng, and Tran given
in 2019, for certain tuple $x=(a_1, \dots, a_n) \in \mathcal{A}^n$, we give a
method to compute a positive element $c_x$ in the C*-algebra $\mathcal{A}$ such
that the equality
  \begin{align*} \sum_{i=1}^{n}(a_ia_i^*)^\frac{1}{2}=c_x \sqrt{n}
\left(\sum_{i=1}^{n}a_ia_i^*\right)^\frac{1}{2}. \end{align*} holds. We give an
application for the integral of G. G. Kasparov. We also derive the formula for
the exact constant for the continuous $\ell_1-\ell_2$ inequality.
