McShane's identity

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

$\left|\sum _{k=1}^n{a}_k{b}_k\right|\le {\max }_{k=1,\dots ,n}\left|B_k\right|(|a_n|+a_n-a_1),$

and if {a_n} is nonincreasing, it holds that

$\left|\sum _{k=1}^n{a}_k{b}_k\right|\le {\max }_{k=1,\dots ,n}\left|B_k\right|(|a_n|-a_n+a_1),$

where

$B_k=b_1+\cdots +b_k.$

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

$\left|\sum _{k=1}^n{a}_k{b}_k\right|\le {\max }_{k=1,\dots ,n}\left|B_k\right|a_1,$

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

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• Abel's inequality in Encyclopedia of Mathematics.

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