### Hölder's Means and Absolute Normalized Norms on $\mathbb{R}^2$

#### Abstract

Recently, Cui and Lu have introduced the constant $H_p(X)\ (p\in\mathbb{R})$ of a Banach space $X$ by using H\"{o}lder'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.

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