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This is the function:

$\displaystyle f(\vec x) = \log \frac{\exp(x_1)}{\sum_{i=1}^n \exp(x_i)} $

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Please do not cross post between mathoverflow and math.stackexchange. – Thomas Rot Jun 22 '11 at 12:27
Irrelevant parts of the question were removed. – Łukasz Lew Jun 22 '11 at 13:54
up vote 4 down vote accepted

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 $\alpha +\beta=1$.

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I don't think this solution works for $g$. Maybe if you come up with some relations between $a_i$'s and $b_i$'s then you may apply something similar. – Beni Bogosel Jun 22 '11 at 13:06

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