Matrix inequalities and majorizations around Hermite–Hadamard’s inequality

Abstract

AbstractWe study the classical Hermite–Hadamard inequality in the matrix setting. This leads to a number of interesting matrix inequalities such as the Schatten p-norm estimates $$ \begin{align*}\left(\|A^q\|_p^p + \|B^q\|_p^p\right)^{1/p} \le \|(xA+(1-x)B)^q\|_p+ \|((1-x)A+xB)^q\|_p, \end{align*} $$ for all positive (semidefinite) $n\times n$ matrices $A,B$ and $0<q,x<1$ . A related decomposition, with the assumption $X^*X+Y^*Y=XX^*+YY^*=I$ , is $$ \begin{align*}(X^*AX+Y^*BY)\oplus (Y^*AY+X^*BX) =\frac{1}{2n}\sum_{k=1}^{2n} U_k (A\oplus B)U_k^*, \end{align*} $$ for some family of $2n\times 2n$ unitary matrices $U_k$ . This is a majorization which is obtained by using the Hansen–Pedersen trace inequality.