Loewner-monotonicity classification for differing parameter sequences

Determine whether the classification of continuously differentiable functions that are entrywise monotone in the Loewner order on each fixed-q Green family remains valid when the two Green matrices being compared have different parameter sequences q.

Background

The paper classifies continuously differentiable functions that satisfy the implication A\succeq B\Rightarrow f[A]\succeq f[B] when A and B belong to the same convex Green family obtained by fixing the parameter sequence q. The resulting functions are characterized through representations of their derivatives as positive mixtures of nonnegative real powers.

The conclusion explicitly identifies the unresolved extension in which the matrices A and B have different q-sequences. Establishing whether the same representation remains necessary and sufficient would extend the Loewner-monotonicity result beyond the fixed-parameter setting.