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.