Bounds for the $p$-angular distance and characterizations of inner product spaces

TL;DR

This paper introduces improved bounds for the $p$-angular distance in normed spaces and provides new characterizations of inner product spaces based on these bounds, enhancing previous results.

Contribution

It offers more accurate bounds for the $p$-angular distance and characterizes inner product spaces through novel inequalities involving this distance.

Findings

New mutual bounds for $p$-angular distance derived

Improved estimates over previous bounds by Dragomir, Hile, and Maligranda

Characterizations of inner product spaces based on $p$-angular distance inequalities

Abstract

Based on a suitable improvement of a triangle inequality, we derive new mutual bounds for $p$-angular distance $\alpha_p[x,y]=\big\Vert \Vert x\Vert^{p-1}x- \Vert y\Vert^{p-1}y\big\Vert$, in a normed linear space $X$. We show that our estimates are more accurate than the previously known upper bounds established by Dragomir, Hile and Maligranda. Next, we give several characterizations of inner product spaces with regard to the $p$-angular distance. In particular, we prove that if $|p|\geq |q|$, $p\neq q$, then $X$ is an inner product space if and only if for every $x,y\in X\setminus \{0\}$, $${\alpha_p[x,y]}\geq \frac{{\|x\|^{p}+\|y\|^{p} }}{\|x\|^{q}+\|y\|^{q} }\alpha_q[x,y].$$