  Let $a$ be a positive element in a unital $C^*$-algebra $\mathfrak{A}$. We
define a semi-norm on $\mathfrak{A}$, which generalizes the $a$-operator
semi-norm and the $a$-numerical radius. We investigate basic properties of this
semi-norm and prove inequalities involving it. Further, we derive new upper and
lower bounds for the $a$-numerical radii of elements in $\mathfrak{A}$. Some
other related results are also discussed.
