Lower and upper bounds for $H$-eigenvalues of even order real symmetric tensors

In this article, we define new classes of tensors called double $\overline{B}$-tensors, quasi-double $\overline{B}$-tensors and establish some of their properties. Using these properties, we construct new regions viz., double $\overline{B}$-intervals and quasi-double $\overline{B}$-intervals, which contain all the $H$-eigenvalues of real even order symmetric tensors. We prove that the double $\overline{B}$-intervals is contained in the quasi-double $\overline{B}$-intervals and quasi-double ${\overline{B}}$-intervals provide supplement information on the Brauer-type eigenvalues inclusion set of tensors. These are analogous to the double $\overline{B}$-intervals of matrices established by J. M. Pe\~na~[On an alternative to Gerschgorin circles and ovals of Cassini, Numer. Math. 95 (2003), no. 2, 337-345.]