Geodesic
Matrix inequalities and majorizations around Hermite–Hadamard’s inequality
Canadian mathematical bulletin, Tome 65 (2022) no. 4, pp. 943-952

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We 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. 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.
DOI : 10.4153/S0008439522000029
Mots-clés : Positive definite matrices, block matrices, convex functions, matrix inequalities 
Bourin, Jean-Christophe; Lee, Eun-Young. Matrix inequalities and majorizations around Hermite–Hadamard’s inequality. Canadian mathematical bulletin, Tome 65 (2022) no. 4, pp. 943-952. doi: 10.4153/S0008439522000029
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     title = {Matrix inequalities and majorizations around {Hermite{\textendash}Hadamard{\textquoteright}s} inequality},
     journal = {Canadian mathematical bulletin},
     pages = {943--952},
     year = {2022},
     volume = {65},
     number = {4},
     doi = {10.4153/S0008439522000029},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008439522000029/}
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