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

2020/10/22 by Mario Krnić, Krnic, Mario, Nicuşor Minculete +1
Mathematics · Medicine · #Mathematical Inequalities and Applications #Nonlinear Partial Differential Equations #Hidradenitis Suppurativa and Treatments

paper · pdf · doi:10.48550/arxiv.2010.11814

Abstract

Based on a suitable improvement of a triangle inequality, we derive new\nmutual bounds for p-angular distance \α_p[x,y]= \Vert Vert\nx Vert^p-1x- Vert y Vert^p-1y \Vert, in a normed linear space X. We\nshow that our estimates are more accurate than the previously known upper\nbounds established by Dragomir, Hile and Maligranda. Next, we give several\ncharacterizations of inner product spaces with regard to the p-angular\ndistance. In particular, we prove that if |p|\≥ |q|, p\≠ q, then X is\nan inner product space if and only if for every x,y\∈ X\∖ 0 ,\n
alpha_p[x,y]
geq
frac
|x
|^p+
|y
|^p
|x
|^q+
|y
|^q\n
alpha_q[x,y].\n

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