A Convexity Proof of Hadamard's Inequality

TLDR: Hadamard's inequality was shown to be true in this article, where it was shown that if p is an extended real valued convex function defined on a convex set A of n X n matrices, A,...,Ak E A and a.1...,ak are nonnegative numbers satisfying Eai = 1, then 4(4aAl + + +aikAk) < ao1(Aj) + * +OakO(Ak).

Abstract: Jensen's Inequality states that if p is an extended real valued convex function defined on a convex set A of n X n matrices, A,,... ,Ak E A and a.1... ,ak are nonnegative numbers satisfying Eai = 1, then 4(4aAl + +aikAk) < ao1.(Aj) + * +OakO(Ak). (2) Finally, a Convexity Result of Minkowski states that if H is an n X n positive semidefinite Hermitian matrix, then (with the convention log 0 = oo) log det H is concave, (3) (det H)1/n is concave. (4) (See e.g., Bellman (1970), p. 128, 132, Marcus and Minc (1964), p. 115, or Marshall and Olkin (1979), p. 475, 476.) Note that (4) implies (3) by virtue of the fact that concavity implies logconcavity. To prove Hadamard's Inequality, let