  In this paper, we introduce a new semi-norm of operators on a semi-Hilbertian
space, which generalizes the A-numerical radius and A-operator semi-norm. We
study the basic properties of this semi-norm, including upper and lower bounds
for it. As an application of this new semi-norm, we obtain upper bounds for the
A-numerical radius of semi-Hilbertian space operators, which are sharper than
the earlier ones.
