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 {fn} be a sequence of real numbers such that fn ≥ fn+1 > 0 for n = 1, 2, …, and let {an} be a sequence of real or complex numbers. Then

\left |\sum_{n=1}^m a_n f_n \right | \le Af_1,

where

A=\operatorname{max}\left \lbrace |a_1|,|a_1+a_2|,\dots,|a_1+a_2+\cdots+a_m| \right \rbrace.

[edit] References

Eric W. Weisstein, Abel's inequality at MathWorld.

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