  We establish local characterizations of matrix monotonicity and convexity of
fixed order by giving integral representations connecting the Loewner and Kraus
matrices, previously known to characterize these properties, to respective
Hankel matrices. Our results are new already in the general case of matrix
convexity and our approach significantly simplifies the corresponding work on
matrix monotonicity. We also obtain an extension of the original
characterization for matrix convexity by Kraus, and tighten the relationship
between monotonicity and convexity.

Functional Analysis