The sharp constant in the Mashreghi-Ransford inequality

Source abstract

Let $(a_n)_{n\geq0}$ 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 $β>1$, put $α=\sqrt{β^2-1}$, and suppose that $b_n,c_n=O(β^n)$. 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 $2/\sqrt{3} \leq κ\leq2$. We prove that the optimal constant is indeed $κ=2/\sqrt{3}$. The proof, which uses exponential generating functions, Phragmén-Lindelöf estimates, and Cauchy's formula, is compared to the classical one of Mashreghi and Ransford.