$SDD_1$ tensors and $B_1$-tensors

Strong $\mathcal{H}$-tensors play an important role in the fields of science and engineering. In this paper, we first propose a new subclass of strong $\mathcal{H}$-tensors that we call the class of $SDD_1$ tensors. We also prove that if a tensor is an $SDD_1$ tensor, it is a strong $\mathcal{H}$-tensor. As an application, a sufficient condition for an $SDD_1$ tensor to certify positive definiteness of even-order real symmetric tensors is proposed. Furthermore, we propose a new class of tensors by means of $SDD_1$ tensors, naming it $B_1$-tensors, and show that $B$-tensors are a subclass of it. Meanwhile, some properties of $B_1$-tensor were introduced. The numerical examples demonstrate the validity of our results.