  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.
