Binary entropy of a probability value $p$ is defined as $$h(p)=-p\log_2(p)-(1-p)\log_2(1-p).$$
I have the following function:$$f(p_1,p_2)=\frac{p_1 h(p_2)-p_2 h(p_1)}{p_1-p_2}$$.
I am interested in finding an approximation to this function when $p_1=e^{-1/a}$ and $p_2=e^{-1/b}$. I know that $b/a=k$, where $k$ is a constant.
Can someone help me simplify $f(p_1,p_2)$?
If I assume $p_1$ or $p_2$ to be approximately $0$ or $1$, this is simple. But, that is not valid in my case.
On
This is too long for a comment to @leonbloy after his/her nice answer.
I think that as soon as you define $p_1=p^{1+x}$ and $p_2=p^{1-x}$, and replaced them in $f(p_1,p_2)$, you can directly expand $f(p_1,p_2)$ as a Taylor series around $x=0$ and get $$\log(2) f(p_1,p_2)=-\log(1-p)+\frac{p (2 p-1)}{6} t\Bigg[1+\frac 1{60(2p-1)}\sum_{n=1}^p (-1)^n {\alpha_n}\,t^n+O(t^{p+1})\Bigg]$$ with $$t=\frac{ \log ^2(p)}{(1-p)^2}x^2$$ the first $\alpha_n$ being (you wrote them) $$\alpha_1=8 p^3-47 p^2+28 p-7$$ $$\alpha_2=\frac{32 p^5-171 p^4+802 p^3-458 p^2+186 p-31 }{42}$$ $$\alpha_3=\frac{128 p^7-1039 p^6+2424 p^5-13753 p^4+6512 p^3-3561 p^2+1016 p-127 }{1680}$$
If $k=b/a$ is not very small or very big then I'd parametrize in this way:
$$\begin{align} p_1 &= p^{1+x} \\ p_2 &= p^{1-x} \\ x &= \frac{\log(p_1/p_2)}{\log(p_1 p_2)} = \frac{b-a }{b+a} =\frac{k-1}{k+1} \\ p &= \sqrt{p_1 p_2}=\exp\left(- \frac{a+b}{2ab}\right)= \exp\left(- \frac{1}{a} \frac{1}{1+x}\right) \end{align} $$
Notice that $p$ is somewhere between $p_1$ and $p_2$, and if (say) $\frac{1}{9}<k<9$ then $-0.8<x<0.8$.
Doing a Taylor expasion of $f$ around $x=0$, we get the approximation
$$ f \approx -\log{\left( 1-p\right) }+\frac{p \left( 2 {{p}}-1\right) {{\log^{2}{(p)}}} }{6 (1-p)^2}{{x}^{2}} $$
(I'm using natural logarithms here, entropy in nats - to get it in bits just divide everything by $\log(2)$)
The approximation seems quite good, assuming $|x|$ is not very near $1$ and $p$ is not too small. Here's a graph for three values of $p$ and $x\in [-0,8,0.8]$. The dashed lines correspond to the approximation.
If you need more precision you can add the next term, which is
$$\frac{p\, \left(7-28p +47 p^2 - 8 p^3 \right) {{\log^4{(p)}}}}{360 {{\left(1-p\right) }^{4}}} x^4$$