Mashreghi–Ransford inequality - Alchetron, the free social encyclopedia

In Mathematics, the Mashreghi–Ransford inequality is a bound on the growth rate of certain sequences. It is named after J. Mashreghi and T. Ransford.

Let ( a n ) n ≥ 0 be a sequence of complex numbers, and let

b n = ∑ k = 0 n ( n k ) a k , ( n ≥ 0 ) ,

and

c n = ∑ k = 0 n ( − 1 ) k ( n k ) a k , ( n ≥ 0 ) .

We remind that the binomial coefficients are defined by

( n k ) = n ! k ! ( n − k ) ! .

Assume that, for some β > 1 , we have b n = O ( β n ) and c n = O ( β n ) as n → ∞ . Then

a n = O ( α n ) , as n → ∞ ,

where α = β 2 − 1 .

Moreover, there is a universal constant κ such that

( lim sup n → ∞ | a n | α n ) ≤ κ ( lim sup n → ∞ | b n | β n ) 1 2 ( lim sup n → ∞ | c n | β n ) 1 2 .

The precise value of κ is unknown. However, it is known that

2 3 ≤ κ ≤ 2.

References

Mashreghi–Ransford inequality Wikipedia

(Text) CC BY-SA