Some Hadamard product inequalities for accretive matrices

Abstract: In this paper, we obtain some new matrix inequalities involving Hadamard product. Also some Hadamard product inequalities for accretive matrices involving the matrix means, positive unital linear maps and matrix concave functions are investigated. Among other results, it is shown that if $A, B, C, D$ are $n\times n$ positive definite matrices, then \begin{equation*} \left(\alpha A+\beta B\right)^r\circ\left(\alpha C+\beta D\right)^{1-r}\leq \alpha\left(A^r\circ C^{1-r}\right)+\beta\left(B^r\circ D^{1-r}\right), \end{equation*} where $r \in (-1, 0) \cup (1, 2)$ and $" \circ "$ stands for the Hadamard product.