Function-Theoretic Operator Norm Inequalities: A Kosaki-type Generalization to Symmetric Probability Weights

Operator norm comparisons play a fundamental role in matrix analysis, yet existing proofs often depend on operator means or interpolation techniques. This study provides a function-theoretic approach to operator norm inequalities. It also extends the classical two-term Heinz comparison to multi-term averages with arbitrary symmetric probability weights. Our approach translates each operator norm comparison into a scalar condition. The condition is derived from functional calculus for the left and right multiplication operators. We examine positive-definiteness and infinite divisibility through Fourier-measure representations. We also use elementary closure properties. For positive operators and any unitarily invariant norm, the two-term Heinz symmetrization is dominated by the binomial average when the exponent differs from one-half by at most one divided by twice the number of terms. For general symmetric probability weights, domination occurs exactly when the exponent lies within a specific threshold. This threshold equals the smallest positive distance from the midpoint to any index carrying nonzero weight. The proposed function-theoretic framework yields necessary and sufficient thresholds to unify the binomial and general symmetric cases.