Crawford numbers and numerical radius of some bordered matrices

The generalized Crawford number [Formula: see text] of a certain class of bordered matrices of the type [Formula: see text] is examined. It is observed that [Formula: see text]. Necessary and sufficient conditions to determine the attainment of the equality [Formula: see text] are given separately which extend the results of [H.-L. Gau, K.-Z. Wang and P. Y. Wu, Crawford numbers of companion matrices, Oper. Matrices 10(4) (2016) 863–879]. Furthermore, a lower bound for the numerical radius of [Formula: see text] (i.e., the case when [Formula: see text]) is provided, which generalizes the result presented in [H.-L. Gau and P. Y. Wu, Lower bounds for the numerical radius, Oper. Matrices 11(4) (2017) 999–1014].