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: Ifand are sequences of real or complex numbers, it holds that