The sharp constant in the Mashreghi-Ransford inequality

Abstract

Let $(a_n{)}_{n\ge 0}$ be a sequence of complex numbers, and define \[ b_n=\sum_{k=0}^n \binom{n}{k} a_k, \qquad c_n=\sum_{k=0}^n \binom{n}{k}(-1)^{n-k}a_k. \] Let $\beta >1$, put $\alpha =\sqrt{{\beta }^2-1}$, and suppose that $b_n,c_n=O({\beta }^n)$. Mashreghi and Ransford proved that \[ \limsup_{n\to\infty}\frac{|a_n|}{α^n} \leq κ\left(\limsup_{n\to\infty}\frac{|b_n|}{β^n}\right)^{\!1/2}\! \left(\limsup_{n\to\infty}\frac{|c_n|}{β^n}\right)^{\!1/2} \] with a universal constant satisfying $2/\sqrt{3}\le \kappa \le 2$. We prove that the optimal constant is indeed $\kappa =2/\sqrt{3}$. The proof, which uses exponential generating functions, Phragmén-Lindelöf estimates, and Cauchy's formula, is compared to the classical one of Mashreghi and Ransford.