I have a function which I am trying to simplify given by $f(K)= \mathrm{h}_{2}\left(E\left[e^{-\frac{\tau^{2}}{K A^{2} \sigma^{2}+\sigma_{w}^{2}}}\right]\right)-\mathrm{E}\left(\mathrm{h}_{2}\left[e^{-\frac{\tau^{2}}{K A^{2} \sigma^{2}+\sigma_{w}^{2}}}\right]\right)$.
Here $K$ is a Binomial RV with parameters $(n,q)$. Also $h_2(p)=-p\log p-(1-p) \log (1-p)$ denotes the binary entropy function.
The parameters $\tau^2, A^2, \sigma_w^2$ and $\sigma^2$ are positive constants. Moreover, $E(.)$ denote the expectation operator.
I realize that $f(K)$ looks like Jensen's inequality gap. Can someone help me simplify $f(K)?$
If it is not possible, I am ok with a good upper bound. I am assuming Jensen's gap can be of some help, but I can't find how.
If even this is not possible, can someone help me identify which value of $q$ maximize $f(K)$ in terms of the other parameters?
In this paper https://ieeexplore.ieee.org/document/9414497 authors propose an upper bound using the concavity of $h_2(q)$ and the fact than the derivative is $h_2^\prime(q)=-\text{logit}(q)$: $$h_2\left(\mathbb{E}\left[Y\right]\right)-\mathbb{E}\left[h_2(Y)\right]=\mathbb{E}\left[h_2\left(\mathbb{E}\left[Y\right]\right)-h_2(Y)\right]\leq\mathbb{E}[(Y-\mathbb{E}[Y])\text{logit}(Y)]$$ In your problem, it is: $$\mathbb{E}\left[\left(e^{-\frac{\tau^2}{KA^2\sigma^2+\sigma_w^2}}-\mathbb{E}\left[e^{-\frac{\tau^2}{KA^2\sigma^2+\sigma_w^2}}\right]\right)\left(-\frac{\tau^2}{KA^2\sigma^2+\sigma_w^2}-\log\left(1-e^{-\frac{\tau^2}{KA^2\sigma^2+\sigma_w^2}}\right)\right)\right]$$ Hope this can help you.