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

Abstract

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.

5 pages

References in corpus (2)

• The geometric mean is a Bernstein function
• A new proof of the geometric-arithmetic mean inequality by Cauchy's integral formula

Cited by in corpus (5)

• A logarithmically completely monotonic function involving the ratio of gamma functions
• The geometric mean is a Bernstein function
• A new proof of the geometric-arithmetic mean inequality by Cauchy's integral formula
• Some Bernstein functions and integral representations concerning harmonic and geometric means
• A ratio of many gamma functions and its properties with applications