Carleman's inequality

The Carleman 's inequality, named after the Swedish mathematician Torsten Carleman, is an elementary inequality of analysis. It says that a number of geometric means of a sequence is bounded by a constant multiple of the row from the top. More specifically, it states that the Euler number is the smallest constant that satisfies as many times this barrier.

The Carleman inequality was first published in 1923 by Torsten Carleman.

Set

Statement

Let be a sequence of real, non- negative numbers. Denote the Euler number. Then:

Here, the smallest number that satisfies this statement.

Evidence

Because is ( telescopic sum )

And follows

And this is after the AM- GM inequality

Variants

For a function with the following continuous variant of Carleman 's inequality holds: