Mashreghi–Ransford inequality explained

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} be a sequence of complex numbers, and letb _{n} = \sum ^{n} _{k =0} {n\choose k} a _{k,} (n \geq 0 ),

and

c _{n} = \sum ^{n} _{k =0} ( -1 ) ^{k} {n\choose k} a _{k,} (n \geq 0 ).

Here the binomial coefficients are defined by

{n\choose k} = (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 inf ty . Then Mashreghi-Ransford showed thata _{n} = O(\alpha ^{n)} , as n \to inf ty ,

where

\alpha = \sqrt{\beta ^{2-1}.} Moreover, there is a universal constant \kappa such that\left( \limsup _{n} (|a _{n|})/(\alpha ^{n}) \right) \leq \kappa \left( \limsup _{n} (|b _{n|})/(\beta ^{n}) \right) ^{(1)/(2)} \left( \limsup _{n} (|c _{n|})/(\beta ^{n}) \right) ^{(1)/(2)} .

The precise value of

\kappa is still unknown. However, it is known that(2)/(\sqrt{ 3) }\leq \kappa \leq 2.

References

• Mashreghi . J. . Ransford . T. . Binomial sums and functions of exponential type . Bull. London Math. Soc. . 37 . 1 . 15–24 . 2005 . 10.1112/S0024609304003625 . 122766740 . .