# History of Arithmetic and Geometric Inequality

I tried to find up the history of (Rogers-)Hölder Inequality. In wikipedia, I found L. J. Rogers's origin paper, saying how to extent the "well known" AM-GM (arithmetic mean-geometric mean) inequality and get more another ones (An extension of a certain theorem in inequalities). I learned AM-GM inequality just for doing math exercise or Math competition, not really understood its own origin. Although it is said that Greek mathematician had the geometric proof (like this Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality) , yet I wonder:

• a) Why did they investigate this and did they know the general case $\frac{a_1+\cdots + a_n}{n}\geq \sqrt[n]{a_1\cdots a_n}$?
• b) Is it used frequently analytically after the emergence of Calculus?
• c) How did it develop from ancient ages to Rogers's ages (1888)?

This would help me solve "the mystery of invention of Hölder and Young inequality" (the final step) in the course of Real-Analysis.