Let $a=(a_1,a_2,...c,a_n)$ for $n\in\mathbb{N}$ be a given sequence ofpositive numbers. In the paper, the authors establish, by using Cauchy'sintegral formula in the theory of complex functions, an integral representationof 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 anew proof of the well known GA mean inequality.
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机译：假设$ n \ in \ mathbb {N} $中的$ a =（a_1，a_2，... c，a_n）$是给定的正数序列。在本文中，作者通过在复杂函数理论中使用柯西的积分公式，建立了几何均值{方程*} 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 \}] $，然后提供众所周知的GA平均不等式的新证明。
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