Abel’s Inequality, named after 19th Century Norwegian mathematician Niels Abel, is an estimate for the sum of products of two numbers.
Let’s ak and bk are two sets of numbers. First, let’s look at the numbers bk and say that the absolute value of all sums Bk = b1 + … + bk , k = 1, …, n are less than or equal to another number, B. Now consider ak. If either of the following are true:
- ai ≥ ai + 1 or
- ai ≤ ai + 1, i = 1, 2, …, n – 1
If the set of numbers ak 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 , stating that if g is monotone, then
is bounded above by:
For example, if, for all values of n,
A > u1 + u2 + … + un > B,
Where ui are real-valued quantities, and if a1, a2, …, an are a series of positive quantities that constantly decrease as n increases, then:
a1A > a1 u1 + a2 u2 + … an un > a1B.
It can also be stated as 
Let z1, …, zn ∈ ℂ, and let Sk = z1 + … + zk, for all k. Then, for each sequence of real numbers a1 ≥ a2 ≥ … ≥ an ≥ 0:
Proof of Abel’s Inequality
We have an = sn − sn − 1, so :
Every |sn| ≤ M and bn − bn + 1 is positive, so
|sn(bn − bn+1)| ≤ M(bn − bn + 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).
 Abel inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Abel_inequality&oldid=34411
 Wolf, J. Sets Whose Difference Set is Square-Free. Retrieved July 3, 2021 from: https://www.cs.umd.edu/~gasarch/TOPICS/vdw/wolfsq.pdf
 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
 Azoff, E. (2010). Sequences and Series. Retrieved July 3, 2021 from: http://alpha.math.uga.edu/~azoff/courses/3100sp10.pdf