Differential analysis of matrix convex functions

Abstract

We analyze matrix convex functions of a fixed order defined on a real interval by differential methods as opposed to the characterization in terms of divided differences given by Kraus. We obtain for each order conditions for matrix convexity which are necessary and locally sufficient, and they allow us to prove the existence of gaps between classes of matrix convex functions of successive orders, and to give explicit examples of the type of functions contained in each of these gaps. The given conditions are shown to be also globally sufficient for matrix convexity of order two. We finally introduce a fractional transformation which connects the set of matrix monotone functions of each order n with the set of matrix convex functions of order n+1.

References in corpus (2)

• Gaps between classes of matrix monotone functions
• Operator monotone functions of several variables

Cited by in corpus (10)

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• Positive polynomials in scalar and matrix variables, the spectral theorem and optimization
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• Differential analysis of matrix convex functions II
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• Loewner matrices of matrix convex and monotone functions
• Subdifferentials of convex matrix-valued functions
• Noncommutative Harmonic and Subharmonic Polynomials and other Noncommutative Partial Differential Equations
• Non-Commutative Harmonic and Subharmonic Polynomials
• Structure of the spaces of matrix monotone functions and of matrix convex functions and Jensen's type inequality for operators