Hadamard-type inequalities for k-positive matrices

We establish Hadamard-type inequalities for a class of symmetric matrices called $k$-positive which the $m$-th elementary functions their eigenvalues are positive all $m\leq k$. These arise naturally in study $k$-Hessian equations Partial Differential Equations. For each matrix, we show that sum its principal minors size $k$ is not larger than $k$-th function diagonal entries. The case $k=n$ corresponds to classical Hadamard inequality definite matrices. Some consequences also obtained.