Some inequalities for log-convex functions of selfadjoint operators on quaternionic Hilbert spaces

TL;DR

This work extends Jensen-type operator inequalities to selfadjoint operators on right quaternionic Hilbert spaces by employing continuous log-convex functions and the quaternionic functional calculus. It refines classical Mond–Pečarić, Lah–Ribarič, and Holder–McCarthy inequalities in the quaternionic setting and provides both single-operator and multi-operator versions. The results yield refined and multiplicative bounds involving $f(T)$, $f(m)$, and $f(M)$, with explicit expressions using the spherical spectrum $\sigma_S(T)$. Applications include quaternionic versions of Ky Fan and Lah–Ribarič inequalities, illustrating concrete operator bounds for functions of $T$ and $I-T$ under spectrum constraints. The findings contribute to a deeper understanding of log-convex function-based inequalities in quaternionic operator theory and their potential applications in quantum and mathematical-physics contexts.

Abstract

In this paper, some Jensen's type inequalities between quaternionic bounded selfadjoint operators on quaternionic Hilbert spaces are proved, using a log-convex function. Also, by applying a specific log-convex function, some particular cases of operator inequalities are obtained.

Theorems & Definitions (24)

• Definition 1
• Remark 2
• Remark 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• proof
• Corollary 8
• Proposition 9