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

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

and

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

Here 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 Mashreghi-Ransford showed that

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

where $\alpha =\sqrt{{\beta }^2-1}.$ Moreover, there is a universal constant $\kappa$ such that

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

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

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

References

• Template:Cite journal.