The maximum and minimum value of homogeneous polynomial under different norms via tensors

For any homogeneous polynomial, it can be expressed as the product of a tensor 51 and a vector x, we denote it by P51(x). With the change of the norm of x, the maximum value (resp. the minimum value) of P51(x) is changed. In this paper, by the properties of tensor 51, we study the relationships between the maximum values (resp. minimum values) of P51(x) under different norms of x. We present that the maximum values (resp. the minimum values) of P51(x) at different norms of x always have the same sign. Moreover, the relationship between the magnitudes of the maximum values (resp. the minimum values) of P51(x) at different norms of x are characterized. Further, some inequalities on H-eigenvalues and Z-eigenvalues of tensor 51 are obtained directly. And some applications on definite positive of tensors and hypergraphs are given.