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

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

and

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

Here the binomial coefficients are defined by

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

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

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

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

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

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

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

References

• Mashreghi, J.; Ransford, T. (2005). "Binomial sums and functions of exponential type". Bull. London Math. Soc. 37 (1): 15–24. doi:10.1112/S0024609304003625. S2CID 122766740..