Reference for "Mazur's inequality"

Asked Nov 22 '22 at 15:39

Active Nov 22 '22 at 15:39

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I was looking for some specific inequalities concerning the absolute value, and I found this particular inequality in a question in this forum: Difference of powers inequality

The inequality is the following: $$ 2^{1-p}|x-y|^p\le\big|x|x|^{p-1}-y|y|^{p-1}\big|\le|x-y|(|x|^{p-1}+|y|^{p-1}), \text{ for }p\ge 1 $$ and $x,y \in \mathbb{R}$.

The user who posted this inequality as an answer named it "Mazur's inequality", but I was not able to find some references with that name on google. Are some of you familiar with this inequality? Can you provide a good reference/proof?

asked Nov 22 '22 at 15:39

Jotabeta

1,283

I found Mazur's inequality (as you stated here) in a Chinese book without references. Also, in a blog entry, the Mazur's inequality is collected without references as well. – River Li Nov 23 '22 at 01:24
Also, perhaps related: https://www.sciencedirect.com/science/article/pii/0898122180900383?ref=pdf_download&fr=RR-2&rr=76e624b28a93faf8 – River Li Nov 23 '22 at 01:32
Thank you! Could you share the link or name for the Chinese book? – Jotabeta Nov 23 '22 at 08:16
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https://www.zhihu.com/question/474094278/answer/2015826434 – River Li Nov 23 '22 at 09:45
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The book: Page 135, (3rd Ed.) https://www.doc88.com/p-096371580918.html – River Li Nov 23 '22 at 09:55
Thank you very much! – Jotabeta Nov 23 '22 at 10:05
You are welcome. – River Li Nov 23 '22 at 12:30

Reference for "Mazur's inequality"

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