p -Uniform Convexity and q -Uniform Smoothness of Absolute Normalized Norms on ℂ 2

Abstract

We first prove characterizations of $p$-uniform convexity and $q$-uniform smoothness. We next give a formulation on absolute normalized norms on ${ℂ}^2$. Using these, we present some examples of Banach spaces. One of them is a uniformly convex Banach space which is not $p$-uniformly convex.