Abel's inequality

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.

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 ight | le Af_1,

where

:A=operatorname{max}left lbrace |a_1|,|a_1+a_2|,dots,|a_1+a_2+cdots+a_m| ight brace.

References


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