We define generalized real and imaginary parts of an operator, as well as the generalized w₇, ₆ () numerical radius, which reduces to the t-weighted numerical radius for suitable functions h, g. Usual properties regarding the new numerical radius are shown, as well as various inequalities concerning the ratio between w₇, ₆ () and w (). In the last section, we give operator matrix inequalities, which generalize the standard numerical radius inequalities, and in one case it is shown that the inequality obtained is sharper than the inequality given by Ammar et al. (Kyungpook Math. J. 65 (1): 63–75, 2025, Theorem 2. 13) for specific operator matrices.
Kıttaneh et al. (Thu,) studied this question.