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
be a sequence of complex numbers, and let
![{\displaystyle b_{n}=\sum _{k=0}^{n}{n \choose k}a_{k},\qquad (n\geq 0),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb1234279725cc372f71baaa8be7ddb689cb0b88)
and
![{\displaystyle c_{n}=\sum _{k=0}^{n}(-1)^{k}{n \choose k}a_{k},\qquad (n\geq 0).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6288e9de48085e215f5a9e70be5e76ed647de0e)
Here the binomial coefficients are defined by
![{\displaystyle {n \choose k}={\frac {n!}{k!(n-k)!}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d37c4e453d63aad9ff3c9cee41cb69f5c7e4e48)
Assume that, for some
, we have
and
as
. Then Mashreghi-Ransford showed that
, as
,
where
Moreover, there is a universal constant
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}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a28f1585c7c1f501c0ca6ae68393fb30e8c288eb)
The precise value of
is still unknown. However, it is known that
![{\displaystyle {\frac {2}{\sqrt {3}}}\leq \kappa \leq 2.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14da37362573cc24033b4665ab9809f6c42c7657)