By virtue of Cauchy's integral formula in the theory of complex functions, the authors establish an integral representation for the weighted geometric mean, apply this newly established integral representation to show that the weighted geometric mean is a complete Bernstein function, and find a new proof of the well-known weighted arithmetic-geometric mean inequality. © 2013 Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg.