Let $\mathbf{x} \in R^{n}$ be a strictly positive vector with $x_1,x_2, \cdots, x_n$ being its elements. I have following function $f(\mathbf{x})$, which is the averaging of $\mathbf{x}$ in logarithmic domain,
\begin{equation} f(\mathbf{x}) = \exp \left( \frac{\sum_{i=1}^n \ln(x_i)}{n}\right) \end{equation}
I wonder, if there exists any tight linear lower bound $g(\mathbf{x})$, for $f(\mathbf{x})$ (i.e., $g(\mathbf{x})\leq f(\mathbf{x})~~ \forall \mathbf{x}>0$) ?