  Let $a=(a_1,a_2,...c,a_n)$ for $n\in\mathbb{N}$ be a given sequence of
positive numbers. In the paper, the authors establish, by using Cauchy's
integral formula in the theory of complex functions, an integral representation
of the principal branch of the geometric mean {equation*}
G_n(a+z)=\Biggl[\prod_{k=1}^n(a_k+z)\Biggr]^{1/n} {equation*} for
$z\in\mathbb{C}\setminus(-\infty,-\min\{a_k,1\le k\le n\}]$, and then provide a
new proof of the well known GA mean inequality.
