Assume that $f:{\bf R}^{n}\to{\bf R}_{+}$ is a concave and homogeneous function of degree one, which satisfies $\frac{\partial f(\lambda)}{\partial\lambda_{j}}>0$ for any $\lambda$ in an admissible cone $\Gamma\subset{\bf R}^{n}$. Assume that there is a $\gamma>0$ such that

$$f(\mu)\geq n\gamma^{1/n}(\prod_{j}\mu_{j})^{1/n},\quad\mbox{for all }\mu\in\Gamma_{n}:=\{\lambda\in{\bf R}^{n}:\lambda_{1}>0,\ldots,\lambda_{n}>0\}.$$

(4.1)

Then $f$ satisfies the structural condition (1.4).