logarithmic average inequality is widely circulated in my country. In many simulation questions in my country, this is the background (for example, many senior high school simulation test final questions in Tianjin 2020-2021 academic year are based on the inequality ). It refers to the following inequality:
Of course, there are also deformations based on this inequality, such as:
In my country, questions with these as background appear not only in the answer questions, but also in some multiple-choice questions that judge the size.
is actually not only available in domestic examinations, but also in foreign exams. For example, Cambridge University is in the independent enrollment examination for high school students. In the first session of the 2017 exam, there is the following question:
You can see that the third question needs to prove is the deformation of the logarithmic average inequality. Here is the official answer to this question:
In the second session, there is the following question:
The first question of this question is also the style of logarithmic average inequality after the exchange of elements. Here is also the official answer:
This question tells us that the logarithmic average inequality is related to the Cauchy Schwartz inequality in integrals. Indeed, the logarithmic average can be derived from this inequality, as follows:
logaverage inequality also has obvious physical significance:
finally attached the history of discovery of logarithmic average inequality:
