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This paper concerns matrix “convex” functions of (free) noncommuting variables, \({x = (x_1, \ldots, x_g)}\). It was shown in Helton and McCullough (SIAM J Matrix Anal Appl 25(4):1124–1139, 2004) that a polynomial in \({x}\) which is matrix convex is of degree two or less. We prove a more general result: that a function of \({x}\) that is matrix convex near \({0}\) and also that is “analytic” in some neighborhood of the set of all self-adjoint matrix tuples is in fact a polynomial of degree two or less. More generally, we prove that a function \({F}\) in two classes of noncommuting variables, \({a = (a_1, \ldots, a_{\tilde{g}})}\) and \({x = (x_1, \ldots, x_g)}\) that is both“analytic” and matrix convex in \({x}\) on a “noncommutative open set” in \({a}\) is a polynomial of degree two or less.

Convex Entire Noncommutative Functions are Polynomials of Degree Two or Less