Abel's inequality

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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.

Let {a1, a2,...} be a sequence of real numbers that is either nonincreasing or nondecreasing, and let {b1, b2,...} be a sequence of real or complex numbers. If {an} is nondecreasing, it holds that

and if {an} is nonincreasing, it holds that


In particular, if the sequence {an} is nonincreasing and nonnegative, it follows that

Abel's inequality follows easily from Abel's transformation, which is the discrete version of integration by parts: If {a1, a2,...} and {b1, b2,...} are sequences of real or complex numbers, it holds that