Abel's inequality explained

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 be a sequence of real numbers that is either nonincreasing or nondecreasing, and let be a sequence of real or complex numbers. If is nondecreasing, it holds that

\left |\sum ^{n} _{k =1} a _{k} b _{k} \right | \le \operatorname{max} _{k=1,...,n} |B _{k|} (|a _{n|} + a _{n} - a _{1),} and if is nonincreasing, it holds that\left |\sum ^{n} _{k =1} a _{k} b _{k} \right | \le \operatorname{max} _{k=1,...,n} |B _{k|} (|a _{n|} - a _{n} + a _{1),} whereB _{k} = b _{1+ … +b} _{k.} In particular, if the sequence is nonincreasing and nonnegative, it follows that\left |\sum ^{n} _{k =1} a _{k} b _{k} \right | \le \operatorname{max} _{k=1,...,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 and are sequences of real or complex numbers, it holds that

\sum ^{n} _{k =1} a _{k} b _{k} = a _{n} B _{n} - \sum ^{n -1} _{k =1} B _{k} (a _{k+1} - a _{k).}

References

• • Abel's inequality in Encyclopedia of Mathematics.