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

$c_n = \sum_{k=0}^n (-1)^{k} {n\choose k} a_k, \qquad (n \geq 0).$

We remind that 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

$a_n = O(\alpha^n)$, 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 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". B. London Math. Soc. 37 (01): 15–24..