**Abel’s Inequality**, named after 19th Century Norwegian mathematician Niels Abel, is an **estimate for the sum of products of two numbers. **

Let’s a_{k} and b_{k} are two sets of numbers. First, let’s look at the numbers b_{k} and say that the absolute value of all sums B_{k} = b_{1} + … + b_{k} , k = 1, …, n are less than or equal to another number, B. Now consider a_{k}. If either of the following are true:

- a
_{i}≥ a_{i + 1}or - a
_{i}≤ a_{i + 1, i = 1, 2, …, n – 1}

then [1]:

If the set of numbers a_{k} is non-increasing and non-negative, the inequality simplifies to:

Uses for Abel’s inequality include:

- Giving a bound for the absolute value of the inner product of two vectors.
- Investigating convergence of sums,
- Showing that certain partial sums form a Cauchy sequence.

## Alternative Definition’s of Abel’s Inequality

There are a few different ways to define the inequality.

One way to define it is as an inequality [2], stating that if *g* is monotone, then

is bounded above by:

For example, if, for all values of n,

A > u_{1} + u_{2} + … + u_{n} > B,

Where u_{i} are real-valued quantities, and if a_{1}, a_{2}, …, a_{n} are a series of positive quantities that constantly decrease as n increases, then:

a_{1}A > a_{1} u_{1} + a_{2} u_{2} + … a_{n} u_{n} > a_{1}B.

It can also be stated as [3]

Let z_{1}, …, z_{n} ∈ ℂ, and let S_{k} = z_{1} + … + z_{k}, for all k. Then, for each sequence of real numbers a_{1} ≥ a_{2} ≥ … ≥ a_{n} ≥ 0:

## Proof of Abel’s Inequality

We have a_{n} = s_{n} âˆ’ s_{n} âˆ’ 1, so [4]:

Every |s_{n}| ≤ M and b_{n} âˆ’ b_{n + 1} is positive, so

|s_{n}(b_{n} âˆ’ b_{n+1})| ≤ M(b_{n} âˆ’ b_{n + 1}).

By the triangle inequality*:

*The Triangle Inequality d(x, y) + d(y, z) ≥ g(x, z) states that the third side of a triangle is always less than the sum of its two sides (or equal, in the case that all points are on a straight line).

## References

[1] Abel inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Abel_inequality&oldid=34411

[2] Wolf, J. Sets Whose Difference Set is Square-Free. Retrieved July 3, 2021 from: https://www.cs.umd.edu/~gasarch/TOPICS/vdw/wolfsq.pdf

[3] Ferber, A. (2020). Arbitrary Topics. Retrieved July 3, 2021 from: https://cpb-us-e2.wpmucdn.com/faculty.sites.uci.edu/dist/f/842/files/2020/06/Inequalities.pdf

[4] Azoff, E. (2010). Sequences and Series. Retrieved July 3, 2021 from: http://alpha.math.uga.edu/~azoff/courses/3100sp10.pdf