Let $A$ be a Banach algebra and let $p$ and $q$ be two positive integers. We show that if

$A$ has a left bounded sequential approximate identity $(e_n)_{n\ge1}$ such that ${\rm lim}\,{\rm inf}_{n\to+\infty}\|e^p_n-e^{p+q}_n\| < ({p \over {p+q}})^{p\over q}{q\over{p+q}}$ then

$A$ has a left-bounded sequential identity $(f_n)_{n\ge1}$ such that $f^2_n = f_n$ for $n\ge1$. A simple example shows that the constant $({p\over {p+q}})^{p\over q}{q\over{p+q}}$ is best possible.

This result is based on some algebraic or integral formulae which associate an idempotent to elements of a Banach algebra satisfying some inequalities involving polynomials or entire functions.