Clarkson's type inequalities for positive $l_p$ sequences with $p\ge 2$

TL;DR

This paper extends Clarkson's inequalities to nonnegative sequences in l_p spaces for p ≥ 2, establishing new inequalities and improvements under certain ordering conditions.

Contribution

It proves a new family of inequalities for nonnegative l_p sequences with p ≥ 2, generalizing Clarkson's inequalities and providing improvements when sequences are ordered.

Findings

Established a generalized Clarkson's inequality for p ≥ 2.

Proved an improved inequality under sequence ordering conditions.

Extended results to nonnegative functions in L_p spaces.

Abstract

For a fixed $1\le p<+\infty$ denote by $\Vert\cdot\Vert_p$ the usual norm in the space $l_p$ (or $L_p$). In this paper we prove that for all real numbers $p$ and $q$ such that $2\le p\le q$ holds $$ 2(\Vert x\Vert_p^q+\Vert y\Vert_p^q)\le \Vert x+y\Vert_p^q +\Vert x-y\Vert_p^q $$ for all nonnegative sequences $x=\{x_n\},y=\{y_n\}$ in $l_p$ (or nonnegative functions $x,y$ in $L_p$). Note that the above inequality with $p=q\ge 2$ reduces to the well known Clarkson's inequality. If in addition, holds $x_i\ge y_i$ for each $i=1,2,...$ (or $x\ge y$ a.e. in $L_p$), then we establish an improvement of the above inequality.