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


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