Inequality of arithmetic and geometric means

Further, we know that the two sides are equal exactly when all the terms of the mean are equal:

Taking antilogs of the far left and far right sides, we have the AM–GM inequality.

For the following proof we apply mathematical induction and only well-known rules of arithmetic.

This case is significantly more complex, and we divide it into subcases.

The following proof uses mathematical induction and some basic differential calculus.

This conjectured inequality was shown by Stephen Drury in 2012. Indeed, he proved

S.W. Drury, On a question of Bhatia and Kittaneh, Linear Algebra Appl. 437 (2012) 1955–1960.

Other generalizations of the inequality of arithmetic and geometric means include: