(I have moved this question from Stackexchange).
Given the operators $\{A_j\}$ in a Banach algebra and a positive integer $p$, let \begin{equation} g=\exp\left(\frac{1}{n}\sum_{j=1}^p A_j\right)\quad\text{and}\quad h=e^{A_1 / n}\dots e^{A_p / n}. \end{equation} Then prove that \begin{align} ||g-h||&\leq||h||\cdot||gh^{-1} -1||\\ &\leq||h||\left\{\exp\left(\frac{2}{n}\sum_{j=1}^p ||A_j || \right)-\left(1+\frac{2}{n}\sum_{j=1}^p ||A_j ||\right)\right\}. \end{align}
This is taken from the proof of Theorem $3$ from this paper, but I cannot see how to get to the second line of the equation above.
