# Mashreghi–Ransford inequality

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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.

Mathematics includes the study of such topics as quantity, structure, space, and change.

Thomas Ransford Ph.D. Sc.D is a British-born Canadian mathematician, known for his research in spectral theory and complex analysis. He holds a Canada Research Chair in mathematics at Université Laval.

Let ${\displaystyle (a_{n})_{n\geq 0}}$ be a sequence of complex numbers, and let

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a solution of the equation x2 = −1. Because no real number satisfies this equation, i is called an imaginary number. For the complex number a + bi, a is called the real part, and b is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, and are fundamental in many aspects of the scientific description of the natural world.

${\displaystyle b_{n}=\sum _{k=0}^{n}{n \choose k}a_{k},\qquad (n\geq 0),}$

and

${\displaystyle c_{n}=\sum _{k=0}^{n}(-1)^{k}{n \choose k}a_{k},\qquad (n\geq 0).}$

We remind that the binomial coefficients are defined by

${\displaystyle {n \choose k}={\frac {n!}{k!(n-k)!}}.}$

Assume that, for some ${\displaystyle \beta >1}$, we have ${\displaystyle b_{n}=O(\beta ^{n})}$ and ${\displaystyle c_{n}=O(\beta ^{n})}$ as ${\displaystyle n\to \infty }$. Then

${\displaystyle a_{n}=O(\alpha ^{n})}$, as ${\displaystyle n\to \infty }$,

where ${\displaystyle \alpha ={\sqrt {\beta ^{2}-1}}.}$

Moreover, there is a universal constant ${\displaystyle \kappa }$ such that

${\displaystyle \left(\limsup _{n\to \infty }{\frac {|a_{n}|}{\alpha ^{n}}}\right)\leq \kappa \,\left(\limsup _{n\to \infty }{\frac {|b_{n}|}{\beta ^{n}}}\right)^{\frac {1}{2}}\left(\limsup _{n\to \infty }{\frac {|c_{n}|}{\beta ^{n}}}\right)^{\frac {1}{2}}.}$

The precise value of ${\displaystyle \kappa }$ is unknown. However, it is known that

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

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