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 {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 {a_n} is nondecreasing, it holds that

$\left |\sum_{k=1}^n a_k b_k \right | \le \operatorname{max}_{k=1,\dots,n} |B_k| (|a_n| + a_n - a_1),$

and if {a_n} is nonincreasing, it holds that

$\left |\sum_{k=1}^n a_k b_k \right | \le \operatorname{max}_{k=1,\dots,n} |B_k| (|a_n| - a_n + a_1),$

where

$B_k =b_1+\cdots+b_k.$

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

$\left |\sum_{k=1}^n a_k b_k \right | \le \operatorname{max}_{k=1,\dots,n} |B_k| a_1,$

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

$\sum_{k=1}^n a_k b_k = a_n B_n - \sum_{k=1}^{n-1} B_k (a_{k+1} - a_k).$

References

• Weisstein, Eric W., "Abel's inequality", MathWorld.
• Abel's inequality in Encyclopedia of Mathematics.