mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, Abel's inequality, named after

Niels Henrik Abel Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solvin ...

, supplies a simple bound on the absolute value of the

inner product In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...

of two vectors in an important special case.

Mathematical description

Let be a sequence of

real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...

s that is either nonincreasing or nondecreasing, and let be a sequence of real or

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

s. If is nondecreasing, it holds that :

\left , \sum_^n a_k b_k \right ,  \le \operatorname_ , B_k,  (, a_n,  + a_n - a_1),

and if is nonincreasing, it holds that :

\left , \sum_^n a_k b_k \right ,  \le \operatorname_ , B_k,  (, a_n,  - a_n + a_1),

where :

B_k =b_1+\cdots+b_k.

In particular, if the sequence is nonincreasing and nonnegative, it follows that :

\left , \sum_^n a_k b_k \right ,  \le \operatorname_ , 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 In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...

: If and are sequences of real or complex numbers, it holds that :

\sum_^n a_k b_k = a_n B_n - \sum_^ B_k (a_ - a_k).

References

* {{mathworld, title=Abel's inequality, urlname=AbelsInequality *
''Abel's inequality''
' in ''

Encyclopedia of Mathematics The ''Encyclopedia of Mathematics'' (also ''EOM'' and formerly ''Encyclopaedia of Mathematics'') is a large reference work in mathematics. Overview The 2002 version contains more than 8,000 entries covering most areas of mathematics at a graduat ...

''. Inequalities Niels Henrik Abel