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 [math]\displaystyle{ (a_n)_{n \geq 0} }[/math] be a sequence of complex numbers, and let

[math]\displaystyle{ b_n = \sum_{k=0}^n {n\choose k} a_k, \qquad (n \geq 0), }[/math]

and

[math]\displaystyle{ c_n = \sum_{k=0}^n (-1)^{k} {n\choose k} a_k, \qquad (n \geq 0). }[/math]

Here the binomial coefficients are defined by

[math]\displaystyle{ {n\choose k} = \frac{n!}{k! (n-k)!}. }[/math]

Assume that, for some [math]\displaystyle{ \beta\gt 1 }[/math], we have [math]\displaystyle{ b_n = O(\beta^n) }[/math] and [math]\displaystyle{ c_n = O(\beta^n) }[/math] as [math]\displaystyle{ n \to \infty }[/math]. Then Mashreghi-Ransford showed that

[math]\displaystyle{ a_n = O(\alpha^n) }[/math], as [math]\displaystyle{ n \to \infty }[/math],

where [math]\displaystyle{ \alpha=\sqrt{\beta^2-1}. }[/math] Moreover, there is a universal constant [math]\displaystyle{ \kappa }[/math] such that

[math]\displaystyle{ \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}}. }[/math]

The precise value of [math]\displaystyle{ \kappa }[/math] is still unknown. However, it is known that

[math]\displaystyle{ \frac{2}{\sqrt{3}}\leq \kappa \leq 2. }[/math]

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. http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=277266. .