Mashreghi–Ransford inequality

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

${\displaystyle b_{n}=\sum _{k=0}^{n}{n \choose k}a_{k},\qquad (n\geq 0),}$

and

${\displaystyle c_{n}=\sum _{k=0}^{n}(-1)^{k}{n \choose k}a_{k},\qquad (n\geq 0).}$

We remind that the binomial coefficients are defined by

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

Assume that, for some ${\displaystyle \beta >1}$, we have ${\displaystyle b_{n}=O(\beta ^{n})}$ and ${\displaystyle c_{n}=O(\beta ^{n})}$ as ${\displaystyle n\to \infty }$. Then

${\displaystyle a_{n}=O(\alpha ^{n})}$, as ${\displaystyle n\to \infty }$,

where ${\displaystyle \alpha ={\sqrt {\beta ^{2}-1}}.}$

Moreover, there is a universal constant ${\displaystyle \kappa }$ such that

${\displaystyle \left(\limsup _{n\to \infty }{\frac {|a_{n}|}{\alpha ^{n}}}\right)\leq \kappa \,\left(\limsup _{n\to \infty }{\frac {|b_{n}|}{\beta ^{n}}}\right)^{\frac {1}{2}}\left(\limsup _{n\to \infty }{\frac {|c_{n}|}{\beta ^{n}}}\right)^{\frac {1}{2}}.}$

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

${\displaystyle {\frac {2}{\sqrt {3}}}\leq \kappa \leq 2.}$