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". Bull. London Math. Soc. 37 (01): 15–24. .

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