New Estimates on the bounds of Brunel's operator.

We study the coefficients of the Taylor series expansion of powers of the function $\psi(x)=\frac{1-\sqrt{1-x}}{x}$, where Brunel's operator $A$ is defined as $\psi(T)$. The operator $A$ was shown to map positive mean-bounded (and power-bounded) operators to positive power-bounded operators. We provide specific details of results announced by A. Brunel and R. Emilion in \cite{Brunel}. In particular, we sharpen an estimate to prove that $\sup_{n\in\mathbb{N}} \|n(A^n-A^{n+1})\| < \infty$. We also provide a proof of the claim that $A$ maps mean-bounded operators to power-bounded operators. Additionally, we prove several new estimates regarding the Taylor series coefficients of $\psi^n$ for $n\in\mathbb{N}$, and derive a closed-form expression for a related summation.