This is the function:
$\displaystyle f(\vec x) = \log \frac{\exp(x_1)}{\sum_{i=1}^n \exp(x_i)} $
This is the function:
$\displaystyle f(\vec x) = \log \frac{\exp(x_1)}{\sum_{i=1}^n \exp(x_i)} $
Pick arbitrary $x,y$ and $\alpha,\beta \in (0,1),\ \alpha+\beta=1$. You want to prove that $f(\alpha x +\beta y) \geq \alpha f(x)+\beta f(y)$. This is equivalent to
$ \log \frac{e^{\alpha x_1+\beta y_1}}{\sum e^{\alpha x_i+\beta y_i}} \geq \log \frac{e^{\alpha x_1+ \beta y_1}}{(\sum e^{x_i})^\alpha (\sum e^{y_i})^\beta} $
and equivalently
$ \sum_{i=1}^n e^{\alpha x_i+\beta y_i}\leq (\sum_{i=1}^n e^{x_i})^\alpha (\sum_{i=1}^n e^{y_i})^\beta $
which is exactly Holder's inequality in the discrete case (note that $\alpha +\beta=1$.