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 {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

$\left|\sum _{k=1}^{n}a_{k}b_{k}\right|\leq \operatorname {max} _{k=1,\dots ,n}|B_{k}|(|a_{n}|+a_{n}-a_{1}),$

and if {a_n} is nonincreasing, it holds that

$\left|\sum _{k=1}^{n}a_{k}b_{k}\right|\leq \operatorname {max} _{k=1,\dots ,n}|B_{k}|(|a_{n}|-a_{n}+a_{1}),$

where

$B_{k}=b_{1}+\cdots +b_{k}.$

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

$\left|\sum _{k=1}^{n}a_{k}b_{k}\right|\leq \operatorname {max} _{k=1,\dots ,n}|B_{k}|a_{1},$

Relation to Abel's transformation

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.