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YL7983 写了： 2023年 11月 11日 16:54 解析数论就是用分析（包括复分析）研究数论。
俺是Littlewood 的第三代(不肖）子孙.
https://www.mathgenealogy.org/

YL7983 写了： 2023年 11月 11日 11:10 三角不等式
柯西-施瓦茨不等式
霍尔德不等式和明可夫斯基不等式
詹森不等式
算术平均数-几何平均数不等式（AM-GM不等式）
杨氏不等式
伯努利不等式

YL7983 写了： 2023年 11月 11日 11:38 Jensen's Inequality is a powerful and versatile tool in mathematics, and it implies or is closely related to several other fundamental inequalities. Among the inequalities I previously mentioned, here's how Jensen's Inequality relates to some of them:

Cauchy-Schwarz Inequality: While Jensen's Inequality doesn't directly imply the Cauchy-Schwarz Inequality, both are used in similar contexts, especially in the study of inner product spaces and Lp spaces. Jensen's Inequality can be used to prove or derive results in these areas, but it doesn't directly imply the Cauchy-Schwarz Inequality.

Triangle Inequality: Jensen's Inequality doesn't directly imply the Triangle Inequality. The Triangle Inequality is more geometric in nature, while Jensen's is more analytical.

Hölder's Inequality and Young's Inequality: Jensen's Inequality is related to both Hölder's and Young's Inequalities in the sense that all these inequalities deal with convexity in some form. However, Jensen's Inequality does not directly imply Hölder's or Young's Inequalities.

Bernoulli's Inequality: Jensen's Inequality does not directly imply Bernoulli's Inequality. While both involve the concept of convex functions, they are used in different contexts.

Markov's Inequality and Chebyshev's Inequality: These inequalities are in the realm of probability theory and are not directly implied by Jensen's Inequality. However, Jensen's Inequality is often used in probability, especially in the context of expected values and convex functions.

Minkowski's Inequality: This inequality is not directly implied by Jensen's Inequality. Minkowski's Inequality is a result in the context of Lp spaces and is more closely related to Hölder's Inequality.

In summary, Jensen's Inequality is a foundational tool in understanding convex functions and has implications in various areas of mathematics. However, it does not directly imply most of the other major inequalities listed, although it can be related to them conceptually or used in similar mathematical contexts.

FoxMe 写了： 2023年 11月 12日 16:06 确实奇怪，这些不等式都能从詹森不等式推出，基本上是基于log, exp, x^p等函数的凹凸性。难道詹森不等式是如此的fundamental？

但是詹森不等式是很晚（20世纪初）才发现的，其他人当时还不知道。有人说证明不等式的常见套路有三：
1. 正性（0阶）；
2. 单调性（1阶）；
3. 凹凸性（2阶）。

那么能不能搞3阶4阶？

FoxMe 写了： 2023年 11月 12日 15:59 赞。有您这样的背景，做点应用研究，比如人工智能，密码学什么的，不是很轻松吗？

YL7983 写了： 2023年 11月 11日 16:54 解析数论就是用分析（包括复分析）研究数论。
俺是Littlewood 的第三代(不肖）子孙.
https://www.mathgenealogy.org/

gmo 写了： 2023年 11月 13日 03:15 华罗庚是你师叔，陈景润潘承洞王元是你师兄，牛逼啊。