Abel's inequality


In mathematics, Abel's inequality, named after Niels Henrik Abel, supplies a simple bound on the absolute value of the inner product of two vectors in an important special case.

Mathematical description

Let be a sequence of real numbers that is either nonincreasing or nondecreasing, and let be a sequence of real or complex numbers.
If is nondecreasing, it holds that
and if is nonincreasing, it holds that
where
In particular, if the sequence is nonincreasing and nonnegative, it follows that

Relation to Abel's transformation">Summation by parts">Abel's transformation

Abel's inequality follows easily from Abel's transformation, which is the discrete version of integration by parts: If
and are sequences of real or complex numbers, it holds that