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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 $(a_n{)}_{n\ge 0}$ be a sequence of complex numbers, and let

$b_n=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})a_k,(n\ge 0),$

and

$c_n=\sum _{k=0}^n(-1{)}^k(\genfrac{}{}{0ex}{}{n}{k})a_k,(n\ge 0).$

We remind that the binomial coefficients are defined by

$(\genfrac{}{}{0ex}{}{n}{k})=\frac{n!}{k!(n-k)!}.$

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

$a_n=O({\alpha }^n)$, as $n\to \mathrm{\infty }$,

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

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

$\left({\mathrm{lim\; sup}}_{n\to \mathrm{\infty }}\frac{\left|a_n\right|}{{\alpha }^n}\right)\le \kappa {\left({\mathrm{lim\; sup}}_{n\to \mathrm{\infty }}\frac{\left|b_n\right|}{{\beta }^n}\right)}^{\frac{1}{2}}{\left({\mathrm{lim\; sup}}_{n\to \mathrm{\infty }}\frac{\left|c_n\right|}{{\beta }^n}\right)}^{\frac{1}{2}}.$

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

$\frac{2}{\sqrt{3}}\le \kappa \le 2.$

References

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