Some Geometric Properties And Constants Of Orlicz Space With Mazur-orlicz F-norm

Geometry of Banach space has important applications in the control theory,fixed point theory,ergodic theory,probability theory and vector analytic function theory.Recall that the monotonicity properties are restrictions of appropriate rotundity properties to the set of couples of comparable elements in the positive cone of a Banach lattice.Consequently,then in many cases,good convex properties can be replaced successfully by respective monotonicity properties.It is well known that monotonicity properties(strict and uniform monotonicity)play analogous role in the best dominated approximation problems in Banach lattices as the respective rotundity properties(strict and uniform rotundity)in the best approximation problems in Banach spaces.Moreover,monotonicity properties are applicable in the ergodic theory,since they provide a tool for estimating a norm.In this paper,based on the paper of 2019 by Yunan Cui,Henryk Hudzik,after removing the continuous condition of generating function,we found some conclusions completely different from those in the Banach spaces we studied in the past,which provided us with a new understanding and enlightenment for studying geometric properties of seminormed linear spaces in the future.This paper mainly studies the monotonicity of this space.First of all,we introduce the development of Orlicz space and monotonicity,and introduce the purpose and significance of this study.Secondly,according to the special properties of the space,some relations between moduli and F-norm of Mazur-Orlicz in Orlicz function space and sequence space are studied.The conditions for the equation(I?(x/?x?)=?x?F)to be true are given,and the criterion for the Orlicz space with Mazur-Orlicz F-norm to be a linear space is given.An example is given to illustrate the essential difference between this space and Banach space.Then,the monotonicity properties of Orlicz function space and Orlicz sequence space given Mazur-Orlicz F-norm are studied:the discriminant rules of strict monotonicity,upper local uniform monotonicity,lower local uniform monotonicity,orthogonal strict monotonicity,orthogonal upper local uniform monotonicity,and orthogonal uniform monotonicity.Finally,by using the monotonicity,an estimate of the orthogonal monotone moduli in LP space with Mazur-Orlicz F-norm is given.