Some Results About The Generalized Von Neumann-Jordan Constant

In this paper, the generalized Von Neumann-Jordan constantCNJ(p)(X) in Banach s-paces are estimated, and the relationship between CNJ(p)(X) and uniformly normal structure of a Banach space X is investigated. This paper is divided the following sections for the demonstration.In the first section, the generalized Von Neumann-Jordan constantCNJ(p)(X) is intro-duced mainly by means of recalling the James constant J(X) and the von Neumann-Jordan constant CNJ(X) and their properties.In the second section, we study the exact values of the von Neumann-Jordan constantCNJ(p)(X) for X being l?-l1, lq - l1, and the regular octagon space. And some new equivalent conditions for uniformly normal structure of a Banach space X are provided by using CNJ(p)(X). Furthermore, uniformly normal structure of a Banach space X are extended.In the last section, when ?·?? is an absolute normalized norm, we consider the gen-eralized von Neumann-Jordan constantCNJ(p)(X) for X being (R2,?·??). In addition, we investigate the relations involving the generalized von Neumann-Jordan constant CNJ(p)(X)of the norms ?·? and ?·? where the convex function ? and ?* are comparable. At last, several concrete examples about the conclusion are listed.