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We establish local characterizations of matrix monotonicity and convexity offixed order by giving integral representations connecting the Loewner and Krausmatrices, previously known to characterize these properties, to respectiveHankel matrices. Our results are new already in the general case of matrixconvexity and our approach significantly simplifies the corresponding work onmatrix monotonicity. We also obtain an extension of the originalcharacterization for matrix convexity by Kraus, and tighten the relationshipbetween monotonicity and convexity.

Local characterizations for the matrix monotonicity and convexity of fixed order