Is it a convex function? $A(\theta) = \log (\sum_{i=1}^n e^{θ⋅f(i)}) - \log (\sum_{i=1}^m e^{θ⋅f(i)})$

Question

$A(\theta) = \log (\sum_{i=1}^n e^{θ⋅f(i)}) - \log (\sum_{i=1}^m e^{θ⋅f(i)})$

Here, $n>m $ and $\forall i, f(i)>0, $. I can prove that $\log (\sum_{i=1}^n e^{θ⋅f(i)})$ is convex and so is true for the other part $\log (\sum_{i=1}^m e^{θ⋅f(i)})$. However, I am finding it very difficult to see if the $A(\theta)$ is convex. Any help is appreciated.

Thanks in advance.

Answer 1

With $(f(1), f(2), f(3)) = (1, 3, 2)$ with $(n, m) = (3, 2)$, it seems to me that

$$ \log(\mathrm{e}^t + \mathrm{e}^{3t} + \mathrm{e}^{2t}) - \log(\mathrm{e}^t + \mathrm{e}^{3t}) = \log \left(1 + \frac{1}{2\cosh t} \right) $$

is not convex. Are you missing some extra conditions?

Answer 2

Yes, some extra conditions are missing. For example, if you plot the curve of

$$f(x)=ln(e^x+e^{x/2}+e^x)-ln(e^x+e^{x/2})$$

you will see that it is first convex then concave.