A new proof of the geometric-arithmetic mean inequality by Cauchy's integral formula

Let $a=(a_1,a_2,\mathrm{.}\mathrm{.}\mathrm{.}c,a_n)$ 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 $z\in \mathbb{C}\setminus (-\mathrm{\infty },-\min \{a_k,1\le k\le n\}]$, and then provide a new proof of the well known GA mean inequality.