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 {an} is nondecreasing, it holds that
| ∑ k = 1 n a k b k | ≤ max k = 1 , … , n | B k | ( | a n | + a n − a 1 ) , {\displaystyle \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 {an} is nonincreasing, it holds that
| ∑ k = 1 n a k b k | ≤ max k = 1 , … , n | B k | ( | a n | − a n + a 1 ) , {\displaystyle \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 + ⋯ + b k . {\displaystyle B_{k}=b_{1}+\cdots +b_{k}.}
In particular, if the sequence {an} is nonincreasing and nonnegative, it follows that
| ∑ k = 1 n a k b k | ≤ max k = 1 , … , n | B k | a 1 , {\displaystyle \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
∑ k = 1 n a k b k = a n B n − ∑ k = 1 n − 1 B k ( a k + 1 − a k ) . {\displaystyle \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.