Abel's inequality [SEP] named after
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 {"a, a..."} be a sequence of real numbers that is either nonincreasing or nondecreasing, and let {"b, b..."} be a sequence of real or complex numbers. 
If {"a"} is nondecreasing, it holds that
and if
{"a"} is nonincreasing, it holds that
where
In particular, if the sequence is nonincreasing and nonnegative, it follows that
Relation to Abel's transformation.
Abel's inequality follows easily from Abel's transformation, which is the discrete version of integration by parts: If
References.
- ""Abel's inequality"" in "Encyclopedia of Mathematics".