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: <math> c_n = \sum_{k=0}^n (-1)^{k} {n\choose k} a_k, \qquad (n \geq 0).</math> : <math> c_n = \sum_{k=0}^n (-1)^{k} {n\choose k} a_k, \qquad (n \geq 0).</math>


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: <math> {n\choose k} = \frac{n!}{k! (n-k)!}.</math> : <math> {n\choose k} = \frac{n!}{k! (n-k)!}.</math>

Revision as of 20:40, 24 October 2022

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 {\displaystyle (a_{n})_{n\geq 0}} be a sequence of complex numbers, and let

b n = ∑ k = 0 n ( n k ) a k , ( n ≥ 0 ) , {\displaystyle b_{n}=\sum _{k=0}^{n}{n \choose k}a_{k},\qquad (n\geq 0),}

and

c n = ∑ k = 0 n ( − 1 ) k ( n k ) a k , ( n ≥ 0 ) . {\displaystyle c_{n}=\sum _{k=0}^{n}(-1)^{k}{n \choose k}a_{k},\qquad (n\geq 0).}

Here the binomial coefficients are defined by

( n k ) = n ! k ! ( n − k ) ! . {\displaystyle {n \choose k}={\frac {n!}{k!(n-k)!}}.}

Assume that, for some β > 1 {\displaystyle \beta >1} , we have b n = O ( β n ) {\displaystyle b_{n}=O(\beta ^{n})} and c n = O ( β n ) {\displaystyle c_{n}=O(\beta ^{n})} as n → ∞ {\displaystyle n\to \infty } . Then Mashreghi-Ransford showed that

a n = O ( α n ) {\displaystyle a_{n}=O(\alpha ^{n})} , as n → ∞ {\displaystyle n\to \infty } ,

where α = β 2 − 1 . {\displaystyle \alpha ={\sqrt {\beta ^{2}-1}}.} Moreover, there is a universal constant κ {\displaystyle \kappa } 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 . {\displaystyle \left(\limsup _{n\to \infty }{\frac {|a_{n}|}{\alpha ^{n}}}\right)\leq \kappa \,\left(\limsup _{n\to \infty }{\frac {|b_{n}|}{\beta ^{n}}}\right)^{\frac {1}{2}}\left(\limsup _{n\to \infty }{\frac {|c_{n}|}{\beta ^{n}}}\right)^{\frac {1}{2}}.}

The precise value of κ {\displaystyle \kappa } is still unknown. However, it is known that

2 3 ≤ κ ≤ 2. {\displaystyle {\frac {2}{\sqrt {3}}}\leq \kappa \leq 2.}

References

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