Some inequalities on the minimum H-eigenvalue for the Fan product and Hadamard product of tensors

For the Fan product and the Hadamard product of tensors, we derive several inequalities for the minimum H-eigenvalue of \(\mathcal{Z}\) -tensors. Using these inequalities, we verify whether the Hadamard product of a nonnegative tensor and a \(\mathcal{Z}\) -tensor is an \(\mathcal{M}\) -tensor and whether it is positive definite. Furthermore, several determinantal properties of the Hadamard product of an \(\mathcal{M}\) -tensor and a nonnegative tensor are established.