Mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the Mashreghi–Ransford inequality is a bound on the growth rate of certain

sequences In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is call ...

. It is named after J. Mashreghi and T. Ransford. Let

(a_n)_

be a sequence of

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

s, and let :

b_n = \sum_^n  a_k, \qquad (n \geq 0),

and :

c_n = \sum_^n (-1)^  a_k, \qquad (n \geq 0).

Here the

binomial coefficients In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the te ...

are defined by :

= \frac.

Assume that, for some

\beta>1

, we have

b_n = O(\beta^n)

and

c_n = O(\beta^n)

n \to \infty

. Then Mashreghi-Ransford showed that :

a_n = O(\alpha^n)

, as

n \to \infty

, where

\alpha=\sqrt.

Moreover, there is a

universal constant A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that cannot be explained by a theory and therefore must be measured experimentally. It is distinct from a mathematical constant, which has a ...

\kappa

such that :

\left( \limsup_ \frac \right) \leq \kappa \, \left( \limsup_ \frac \right)^  \left( \limsup_ \frac \right)^.

The precise value of

\kappa

is still unknown. However, it is known that :

\frac\leq \kappa \leq 2.

References

* . {{DEFAULTSORT:Mashreghi-Ransford inequality Inequalities (mathematics)