An Unexpected AGM-Type Inequality and a New Characterization of Operator Monotone Functions

Key Points

• The aim is to establish a new inequality related to the arithmetic-geometric mean for positive definite matrices and apply it to classify operator monotone functions.
• Developed a new arithmetic-geometric mean type inequality.
• Applied the inequality to characterize operator monotone functions.
• Demonstrated the existence of a new inequality for positive definite matrices.
• Provided a characterization of operator monotone functions using the inequality.