A Boundary Condition for the Arithmetic-Geometric Mean Inequality Under Complex Extension to Mixed-Sign Pairs

Key Points

• The aim is to explore the implications of the AM-GM inequality when applied to complex and mixed-sign pairs.
• Analyzed the classical AM-GM inequality framework.
• Examined the effects of incorporating negative and mixed-sign numbers.
• Developed mathematical proofs to investigate boundary conditions.
• Identified specific conditions under which the AM-GM inequality holds for mixed-sign inputs.
• Showed that certain pairs can violate the traditional AM-GM relationship.