In this paper, we propose an alternative formulation of the weighted A-numerical radius ω (₀, ₕ) (·) for bounded linear operators on semi-Hilbert spaces induced by a positive operator A. Building upon the framework introduced by Gao and Liu Oper. Matrices, 17 (2023), 343-354, we derive several new inequalities that offer sharper bounds and refined characterizations of this semi-norm. These results enrich the existing theory by extending the analytical tools available for studying operator behavior in semi-Hilbertian contexts.
Ren et al. (2025) studied this question.