Matrix inequalities and majorizations around Hermite–Hadamard’s inequality

Abstract We study the classical Hermite–Hadamard inequality in matrix setting. This leads to a number of interesting inequalities such as 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 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^*, some family $2n\times 2n$ unitary $U_k$ majorization which obtained by using Hansen–Pedersen trace inequality.