An Equality Condition of Arhippainen-Müller's Estimate and its Related Problem

J.~Arhippainen and V.~M\"{u}ller showed that $\Vert_1{a + \lambda e} \leq 3\Vert_{a + \lambda e}$ for all $a + \lambda e \in \tilde{A}$, where $\tilde{A}$ is the unitization of an arbitrary non-unital Banach algebra $A$ with regular norm. In this note, we present a necessary and sufficient condition for the equality of this estimate and its related problem.