A remark on the norm of the parallel sum

It is shown that if \(a\!:\!b\) is the parallel sum of the two positive definite elements a and b of a \(C^*\)-algebra, then for any \(s, t\in [0, 1]\),

$$\begin{aligned} \big \Vert a\!:\!b\big \Vert \le \frac{1}{2}\left( \Vert a\Vert \!:\!\Vert b\Vert + \frac{\Vert a\Vert :\Vert b\Vert }{\Vert a\Vert +\Vert b\Vert }\sqrt{\left( \Vert a\Vert -\Vert b\Vert \right) ^2 +4\left\| a^{1-s}b^{t}\right\| \left\| a^{s}b^{1-t}\right\| }\,\right) . \end{aligned}$$

This inequality, which is sharper than the inequality \(\big \Vert a\!:\!b\big \Vert \le \Vert a\Vert \!:\!\Vert b\Vert \), generalizes an earlier related inequality.