optimal $z$-eigenvalue bounds and rank-one approximations

This study explores the $z$-eigenvalue inclusion theorem, focusing on its role in improving the best rank-one approximation for tensors. we propose novel inclusion sets inspired by brauer and brualdi's frameworks, offering sharper bounds on $z$-eigenvalues. these sets are demonstrated to provide more accurate results than existing approaches. additionally, we apply these results to obtain refined estimates of best rank-one approximation, particularly for weakly symmetric nonnegative tensors. the paper includes numerical examples to validate the enhanced bounds presented.