Let $n > 1$ and $\sigma \in \mathbb R_{++} \setminus \{1\}$. Consider the following function \begin{align} f(\sigma) = \frac{\sigma}{n}\left[\frac{1}{n}\sum_{j=1}^n{t_j^{1-\sigma}}\right]^\frac{1}{\sigma-1} - 1 -\sigma \end{align} where $(t_1,\ldots,t_n) \in \mathbb R_{++}^n$.
In want to show that $\lim_{\sigma \to 1}f(\sigma) = 0$ is the unique solution for $f(\sigma) = 0$.
Let us first verify that $\lim_{\sigma \to 1}f(\sigma) = 0$ is true \begin{align} f(\sigma) = 0 \quad \Longrightarrow \quad \sum_{j=1}^n{t_j^{1-\sigma}} = \left(\frac{1+\sigma}{\sigma}\right)^{\sigma-1}n^\sigma. \end{align} The right hand equality is true for $\sigma \to 1$. How do I verify that there cannot be another $\sigma$ verifying $f(\sigma) = 0$?