Let be positive real numbers satisfying Then if are nonnegative real numbers, and equality holds if and only if
Proof 1 (using the weighted AM-GM inequality): Let Then the weighted AM-GM inequality says that but so the left side is and the right side is The statement that equality holds if and only if also follows directly from the statement of the weighted AM-GM inequality.
Proof 2 (using logarithms): It suffices to prove the theorem for The function is concave down, so for all positive and with equality holding if and only if
Now set so and exponentiate both sides.
The case is just the AM-GM inequality for :
Let be a positive integer. Find the minimum value of for positive real numbers
This could be done by AM-GM on and copies of but Young's inequality works as well: divide by to get and note that satisfy which suggests multiplying by : Now let Then so and in fact equality holds when or
As mentioned in the introduction, Young's inequality is essential in the proof of Hölder's inequality; see the wiki for details.
Young's inequality for products is a special case of Young's inequality for increasing functions:
Let be a continuous, increasing function defined for nonnegative real numbers with Suppose are positive real numbers such that is in the domain of and is in the image of Then with equality if and only if
The proof is quite elegant: the picture for is similar. Equality only holds if there is no extra area, which is whenThis is the picture for
Let Then and so and is the of Young's inequality, so we recover Young's inequality:
- Duke, N. Young Inequality. Retrieved July 28, 2011, from https://commons.wikimedia.org/wiki/File:Young_inequality.svg