A Quaternionic Hansen--Pedersen Inequality and its Application to $S_r$-Transforms
We establish a quaternionic Hansen--Pedersen inequality for real operator-convex functions on right quaternionic Hilbert spaces. The proof is based on the continuous real functional calculus for selfadjoint quaternionic operators, a direct-sum functional-calculus identity, and a block-unitary construction. As an application, let \(T = U|T|\) be an injective \(p\)-hyponormal operator. We prove that, for \(0 < r \leq 1/2\), the transform \(S_{r}(T) = U|T|^{r}U\) is \(2p\)-hyponormal when \(0 < p \leq 1/2\) and hyponormal when \(1/2 < p \leq 1\). We also show that \(S_{r}(T)\) preserves log-hyponormality for \(r > 0\). These results illustrate the use of quaternionic Jensen-type operator inequalities in the study of transforms associated with the polar decomposition.