Recently, Cui and Lu have introduced the constant $H_p(X)$ $(p\in\mathbb R)$ of a Banach space $X$ by using Hólder's means. In this paper, we determine and estimate the new constant under the absolute normalized norms on $\mathbb R^2$ by means of their corresponding continuous convex functions on $[0,1]$. Furthermore, the exact values of the constant are calculated in some concrete Banach spaces. In particular, we calculate the precise values of the constants $A_2(X)$ and $T(X)$ and the Gao constant $E(X)$ in these concrete spaces.