Abel’s Inequality: Definition & Proof

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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:

  1. ai ≥ ai + 1 or
  2. ai ≤ ai + 1, i = 1, 2, …, n – 1

then [1]:
abel inequality

If the set of numbers ak is non-increasing and non-negative, the inequality simplifies to:
abel's formula

Uses for Abel’s inequality include:

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
abels inequality 1

is bounded above by:
abels inequality 2

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 [3]
Let z1, …, zn ∈ ℂ, and let Sk = z1 + … + zk, for all k. Then, for each sequence of real numbers a1 ≥ a2 ≥ … ≥ an ≥ 0:
abels inequality 3

Proof of Abel’s Inequality

We have an = sn − sn − 1, so [4]:
proof of abels' inequality

Every |sn| ≤ M and bn − bn + 1 is positive, so
|sn(bn − bn+1)| ≤ M(bn − bn + 1).

By the triangle inequality*:
proof of abels 2

*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).


[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

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