The sharp constant in the Mashreghi-Ransford inequality

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